Dixmier Traces 194 6.1 Introduction

Home , Commutator subspace, Operator topologies, Singular trace

De Gruyter Studies in Mathematics 46

Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Steven Lord Fedor Sukochev Dmitriy Zanin Singular Traces

Theory and Applications

De Gruyter Mathematical Subject Classiﬁcation 2010: 46L51, 47L20, 58B34, 47B06, 47B10, 46B20, 46E30, 46B45, 47G10, 58J42.

ISBN 978-3-11-026250-6 e-ISBN 978-3-11-026255-1

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Printed in Germany www.degruyter.com Preface

This book is dedicated to the memory of Nigel Kalton. Nigel was going to be a co- author but, tragically, he passed away before the text could be started. The book has become a tribute. A tribute to his influence on us, and a tribute to his influence on the area of singular traces. It would not have been written without his inspiration on techniques concerning symmetric norms and quasi-nilpotent operators. A lot of the development was very recent. Singular traces is still an evolving mathematical field. Our book concentrates very much on the functional analysis side, which we feel is heading toward some early form of maturity. It was a sense of consolidation, that we had something worth report- ing, which sponsored the idea of writing a book. We thought that new results discovered with Nigel, based on previous work about symmetric spaces, symmetric functionals and commutator subspaces by Nigel, Peter and Theresa Dodds, Ken Dykema, Thierry Fack, Tadius Figiel, Victor Kaftal, Ben de Pagter, Albrecht Pietsch, Alek- sander Sedaev, Evgenii Semenov, Gary Weiss, Mariusz Wodzicki, and many others, deserved a wider audience amongst operator algebraists, functional analysts, noncommutative geometers and allied mathematical physicists. Despite the patchiness of a young field, what is emerging is that singular traces are as fundamental to the study of functional analysis as the canonical trace. The book is complete, in the limited sense that any investigation can be complete, as a story of the existence of continuous traces on symmetrically normed ideals. It is a complete story about the Calkin correspondence and the Lidskii eigenvalue formula for continuous traces. It is a complete story of the Dixmier trace on the dual of the Macaev ideal. It is a complete story about Connes’ Trace Theorem. We strongly emphasize, however, that this book is not a reference for, nor a snapshot of, the entire field. We have included end notes to each chapter, giving historical background, credit of results, and reference to alternative approaches to the best of our knowledge; we apologize in advance for possible omissions. We hope we have made the topic accessible, as well as displaying what we think to be the vital and interesting features of singular traces on symmetric operator spaces.

The authors acknowledge their families for their invaluable support. Steven Lord and Fedor Sukochev thank their partners, Rebekkah Sparrow and Olga Lopatko. We thank Fritz Gesztesy for the introduction to the publisher and assistance during the publication process, and for his help with the permission to reproduce a tribute from the Nigel Kalton Memorial Website. We thank Anna Tomskova for LaTeX and editing support. vi Preface

We thank Albrecht Pietsch for helping us with historical comments. We thank Jacques Dixmier for his permission to quote from the letter he wrote to the conference “Singu- lar Traces and Their Applications” (Luminy, January 2012). We thank colleagues that have given us invaluable advice, encouragement, and criticism, Alan Carey, Vladimir Chilin, Victor Gayral, John Phillips, Denis Potapov, and Adam Rennie, amongst those not already mentioned, and we acknowledge contributors and co-workers in the ﬁeld, Nurulla Azamov, Daniele Guido, Bruno Iochum, Tommaso Isola, Alexandr Usachev, and Joseph Várilly, amongst many others.

Sydney, Australia, 30 July 2012 Steven Lord Fedor Sukochev Dmitriy Zanin Notations

N set of natural numbers Z set of integers ZC set of non-negative integers R field of real numbers RC set of positive real numbers Rd Euclidean space of dimension d jj Euclidean norm on Rd C field of complex numbers MN .C/ algebra of square N N complex matrices Tr.A/ trace of a matrix A det.A/ determinant of a matrix A L N direct sum of Banach spaces, traces or operators tensor product of von Neumann algebras, traces or operators ? convolution orthocomplement submajorization C uniform submajorization uniform convergence a˛ " a (a˛ # a) increasing (decreasing) net with respect to a partial order Gamma function logC.x/ maxflog jxj,0g H complex separable Hilbert space h, i inner product (complex linear in the first variable) kk norm (usually the vector norm on a Hilbert space) kk1 operator (uniform) norm L.H / algebra of bounded operators on H 1 fengnD0 orthonormal basis of H ˝ one-dimensional operator on H defined by . ˝ /x :Dhx, i 1 identity map on H diag.a/ diagonal operator on H associated with a 2 l1 ŒA, B commutator of operators A and B A adjoint of the operator A jAj absolute value of the operator A viii Notations

A 0 operator A is positive AC (A) positive (negative) part of a self-adjoint operator A sequence space JC positive part of the space J Z.J / center of J J two-sided ideal of compact operators JC positive part of the ideal J Com.J / commutator subspace of J c00 space of eventually vanishing sequences c0 space of converging to zero sequences c space of convergent sequences l1 space of bounded sequences l1 space of summable sequences l2 Hilbert space of square summable sequences lp space of p-summable sequences lp,1 weak-lp sequence space lˆ Orlicz sequence space associated with an Orlicz function ˆ m1,1 Sargent sequence space C00.H / ideal of ﬁnite rank operators on H C0.H / ideal of compact operators on H L1 ideal of trace class operators L2 ideal of Hilbert-Schmidt operators Lp Schatten-von Neumann ideal of compact operators Lp,1 weak-lp ideal of compact operators Lˆ ideal of compact operators associated with lˆ M1,1 Dixmier-Macaev ideal of compact operators .M1,1/0 separable part of M1,1 kkp (standard) norm on lp, Lp or Lp kk1,w quasi-norm on l1,1 or L1,1 Notations ix

Tr.A/ (standard) trace of the trace class operator A Tr!.A/ Dixmier trace (discrete version) of the operator A C.X/ space of continuous functions on a compact Hausdorff topological space X C0.X/ space of continuous functions vanishing at infinity on a locally compact Hausdorff topological space X .X, †, / -finite measure space L1.X/ algebra of (equivalence classes of) bounded functions on X L1.X/ space of (equivalence classes of) integrable functions on X L2.X/ Hilbert space of (equivalence classes of) square integrable functions on X Lp.X/ space of (equivalence classes of) p-integrable functions on X A characteristic function of a measurable set A X trace associated with a measure m Lebesgue measure L0.0, 1/ set of measurable functions on RC [f1g C R L0 .0, 1/ set of positive measurable functions on C [f1g S.0, 1/ set of all (m)-measurable functions on .0, 1/ (on .0, 1/) (S.0, 1/) Lp Lebesgue Lp-space on .0, 1/ or .0, 1/ 0 L1.0, 1/ space of (equivalence classes of) bounded functions vanishing at infinity Mx multiplication operator on L2.X/ for x 2 L1.X/ supp.x/ support of a sequence x, a complex valued function x,oradistri- bution x

M semifinite von Neumann algebra L.H /C (MC) set of positive elements of L.H / (M) Proj.M/ lattice of projections in M _ (^) supremum (infimum) operation (in a lattice) faithful normal semifinite trace on M .M, / von Neumann algebra M equipped with a faithful normal semifinite trace S.M, / set of -measurable operators Lp.M, / noncommutative Lp-space C0.M, / ideal of -compact operators EA spectral measure of the operator A 2 M nA spectral distribution function of the operator A 2 M E symmetric sequence or function space kkE symmetric norm on E E0 separable part of E x Notations

E symmetric ideal of compact operators E.M, / symmetric operator space associated to the pair .M, / kkE norm on E or E.M, / E0 (E.M, /0) separable part of E (E.M, /) E (E.M, /) dual space of E (E.M, /) E.jA/ expectation operator associated with the partition A DE Lin.fx 2 E : x D .x/g/ ZE Lin.fx1 x2 :0 x1, x2 2 E, .x1/ D .x2/g/ ' trace, continuous trace or (fully) symmetric functional L.'/./ symmetric functional on E.M, / associated with a symmetric functional ' on E 'C positive part of a hermitian symmetric functional ' positive concave increasing function on .0, 1/ 0 derivative of m Lorentz sequence space associated with m Köthe dual space to m M Lorentz ideal of compact operators associated with m M Lorentz ideal of compact operators associated with m M Lorentz function space associated with M Köthe dual space to M M .M, / Lorentz operator space associated with M M M . , / Lorentz operator space associated with M M1,1 Lorentz function space associated with .t/ D log.1 C t/ M1,1.M, / Dixmier-Macaev operator ideal associated with .M, / and M1,1 C Cesàro operator M logarithmic mean operator M discrete logarithmic mean operator n./ dilation operator on l1 s./ dilation operator on L1.0, 1/ associated with s>0 Ps exponentiation operator on L1.0, 1/ associated with s>0 lim ordinary limit on c ! dilation invariant extended limit on l1 or L1.0, 1/ extended limit on L1.0, 1/

! Dixmier trace (continuous version) -function associated with ,B -function associated with the operator B and heat kernel function associated with ,B heat kernel function associated with the operator B and Notations xi

@ˇ partial derivative associated with the multi-index ˇ Laplacian on Rd hsi˛ .1 Cjsj2/r=2, s 2 Rd , r 2 R C 1.Rd / set of smooth functions on Rd 1 Rd Rd Cc . / Frechet space of smooth functions on with compact support D0 Rd 1 Rd . / space of distributions (the dual space of Cc . /) S.Rd / Frechet space of Schwartz functions on Rd S0.Rd / space of tempered distributions (the dual space of S.Rd /) d Lmod;1.R / algebra of modulated functions d Lmod;2.R / Banach space of functions whose absolute value squared is a modulated function Lm set of (uniform) pseudo-differential operators of order m KA kernel of a pseudo-differential or Laplacian modulated operator A pA symbol of a pseudo-differential or Laplacian modulated operator A S m set of (uniform) symbols of order m F Fourier transform L1 set of smoothing operators S 1 set of symbols of smoothing operators G1 set of Shubin smoothing operators H r .Rn/ Sobolev Hilbert space of order r 2 R Sd1 unit sphere in Rd ResW noncommutative residue on the set of (compactly based) classical pseudo-differential operators kkmod norm on the space of Laplacian modulated operators Res vector-valued noncommutative residue on the set of (compactly based) Laplacian modulated operators S mod Banach space of symbols of Laplacian modulated operators .X, g/ closed Riemannian manifold X and Riemannian metric g .U , h/ chart of a manifold X C 1.X/ space of smooth functions on X det g determinant of the matrix of coordinates of the metric g Laplace-Beltrami operator TX tangent bundle of the manifold X S X cosphere bundle of the manifold X H r .X/ Sobolev Hilbert space of order r 2 R on the manifold X Lm.X/ set of pseudo-differential operators of order m on the manifold X .‰,g/ pT coordinate-dependent symbol associated with a partition of unity ‰ and metric g .H , D/ unbounded Fredholm module associated with Hilbert space H and self-adjoint operator D xii Notations hDir .1 C D2/r=2, r 2 R ResD vector-valued noncommutative residue associated with .H , D/ Int! noncommutative integral associated with ! Int.A/ noncommutative integral of an operator A which does not depend on ! Tr r-torus ƒ additive group of skew symmetric matrices in Mr .T/ L‚.A/ isospectral deformation of an operator A associated with ‚ 2 ƒ N‚ noncommutative torus Contents

Preface v Introduction 1

I Preliminary Material 1 What is a Singular Trace? 15 1.1 CompactOperators ...... 15 1.2 Calkin Correspondence ...... 22 1.3 Examples of Traces ...... 28 1.3.1 The Canonical Trace ...... 29 1.3.2 TheDixmierTrace ...... 29 1.3.3 Lidskii Formulation of Traces ...... 33 1.4 Notes...... 34 2 Preliminaries on Symmetric Operator Spaces 38 2.1 VonNeumannAlgebras ...... 38 2.2 Semiﬁnite Normal Traces ...... 42 2.3 Generalized Singular Value Function ...... 46 2.4 Calkin Correspondence in the Semiﬁnite Setting ...... 53 2.5 Symmetric Operator Spaces ...... 56 2.6 Examples of Symmetric Operator Spaces ...... 59 2.7 Traces on Symmetric Operator Spaces ...... 68 2.8 Notes...... 70

II General Theory 3 Symmetric Operator Spaces 79 3.1 Introduction ...... 79 3.2 Submajorization in the Finite-dimensional Setting ...... 80 3.3 Hardy–Littlewood(–Polya) Submajorization ...... 83 3.4 UniformSubmajorization ...... 88 3.5 Symmetric Operator Spaces from Symmetric Function Spaces ...... 97 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces ...... 101 3.7 Notes...... 104 xiv Contents

4 Symmetric Functionals 107 4.1 Introduction ...... 107 4.2 Jordan Decomposition of Symmetric Functionals ...... 109 4.3 Lattice Structure on the Set of Symmetric Functionals ...... 114 4.4 LiftingofSymmetricFunctionals...... 117 4.5 Figiel–KaltonTheorem...... 120 4.6 ExistenceofSymmetricFunctionals...... 123 4.7 ExistenceofFullySymmetricFunctionals...... 130 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 133 4.9 SymmetricFunctionalsonSymmetricOperatorSpaces...... 142 4.10HowLargeistheSetofSymmetricFunctionals?...... 146 4.11Notes...... 152 5 Commutator Subspace 153 5.1 Introduction ...... 153 5.2 NormalOperatorsintheCommutatorSubspace...... 155 5.3 NormalOperatorsintheClosedCommutatorSubspace...... 162 5.4 Subharmonic Functions on Matrix Algebras ...... 168 5.5 Quasi-nilpotentOperatorsBelongtotheCommutatorSubspace ..... 173 5.6 DescriptionoftheCommutatorSubspace...... 182

5.7 Commutator Subspace of the Weak Ideal L1,1 ...... 187 5.8 Notes...... 192 6 Dixmier Traces 194 6.1 Introduction ...... 194 6.2 ExtendedLimits...... 196 6.3 DixmierTracesonLorentzIdeals...... 198 6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces . . 203 6.5 Dixmier Traces on Fully Symmetric Ideals of L.H / ...... 206 6.6 RelativelyNormalFunctionals...... 209 6.7 WodzickiRepresentationofDixmierTraces ...... 214 6.8 Notes...... 217 Contents xv

III Traces on Lorentz Ideals 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals 225 7.1 Introduction ...... 225 7.2 Distribution Formulas for Dixmier Traces ...... 226 7.3 Lidskii Formulas for Dixmier Traces ...... 232 7.4 Special Cases and Counterexamples ...... 235 7.5 Diagonal Formulas for Dixmier Traces Fail ...... 241 7.6 Notes...... 242 8 Heat Kernel Formulas and -function Residues 244 8.1 Introduction ...... 244 8.2 HeatKernelFunctionals ...... 246 8.3 FullySymmetricFunctionalsareHeatKernelFunctionals...... 252 8.4 GeneralizedHeatKernelFunctionals...... 256 8.5 ReductionofGeneralizedHeatKernelFunctionals ...... 258 8.6 -functionResidues...... 263 8.7 Not Every Dixmier Trace is a -functionResidue ...... 268 8.8 Notes...... 271 9 Measurability in Lorentz Ideals 272 9.1 Introduction ...... 272 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals ...... 273

9.3 Positive Dixmier Measurable Operators in M1,1 ...... 276 9.4 C -invariantExtendedLimits...... 281 9.5 Positive M -measurableOperators...... 287 9.6 Additional Invariance of Dixmier Traces ...... 291

9.7 Measurable Operators in L1,1 ...... 297 9.8 Notes...... 300

IV Applications to Noncommutative Geometry 10 Preliminaries to the Applications 311

10.1 Summary of Traces on L1,1 and M1,1 ...... 311 10.2 Pseudo-differential Operators and the Noncommutative Residue . . . . . 317 10.3 Pseudo-differential Operators on Manifolds ...... 330 10.4Notes...... 334 xvi Contents

11 Trace Theorems 336 11.1 Introduction ...... 336 11.2 Modulated Operators ...... 339 11.3 Laplacian Modulated Operators and Extension of the NoncommutativeResidue...... 345 11.4 Eigenvalues of Laplacian Modulated Operators ...... 355 11.5 Trace Theorem on Rd ...... 359 11.6TraceTheoremonClosedRiemannianManifolds ...... 362 11.7IntegrationofFunctions...... 372 11.8Notes...... 380 12 Residues and Integrals in Noncommutative Geometry 382 12.1 Introduction ...... 382 12.2TheNoncommutativeResidueinNoncommutativeGeometry...... 385 12.3TheIntegralinNoncommutativeGeometry...... 390 12.4ExampleofIsospectralDeformations...... 396 12.5ExampleoftheNoncommutativeTorus ...... 405 12.6ClassicalLimits...... 411 12.7Notes...... 416 A Operator Results 420 A.1 MatrixResults...... 420 A.2 Operator Inequalities ...... 422 Bibliography 429 Index 445 Introduction

This book concerns traces on symmetric operator spaces1. It details a functional analysis approach to singular traces developed over the last four decades.

What is a Singular Trace? For many years trace theory in the theory of Hilbert spaces has concentrated on the canonical operator trace. Operators on Hilbert spaces are an extension of linear algebra and the operator trace is the extension of the matrix trace. Fundamental observations about the matrix trace extend to the operator trace. For a continuous linear operator A mapping the Hilbert space to itself there is the description of the operator trace as the sum of the diagonal, X1 Tr.A/ D hAen, eni (1) nD0

1 where fengnD0 is an arbitrary orthonormal basis of the Hilbert space. It was probably John von Neumann in 1932 that ﬁrst abstractly formulated the operator trace in this way. There is also the description of the operator trace as the sum of eigenvalues X1 Tr.A/ D .n, A/ (2) nD0 where A is a compact linear operator mapping the Hilbert space to itself and 1 2 f.n, A/gnD0 are the eigenvalues of A. This result was shown by Victor Lidskii in 1959 and the formula bears his name. There can be several candidates for “inﬁnite dimensional matrices” in the Hilbert space theory3. A bounded linear operator A maps the Hilbert space to itself so that

1 Symmetric operator spaces in our sense, which includes symmetrically normed operator ideals, symmetric function spaces and symmetric sequence spaces, should not be confused with operator spaces in the sense of a Banach space with an isometric linear embedding into the set of bounded operators on some Hilbert space. 2 Whenever eigenvalues of compact operators appear we shall always consider them as a sequence of the non-zero eigenvalues, with multiplicity, ordered in such a way that their absolute values decrease. When there are no non-zero eigenvalues then the sequence is the zero sequence. 3 The analogy is to help readers who may not be familiar with functional analysis, but it should not be taken too far as the behavior of linear operators on infinite dimensional Hilbert spaces can be quite distinct to linear transformations of finite dimensional spaces. 2 Introduction kAxkC kxk for some constant C and for every vector x of the Hilbert space. The smallest constant C defines the operator norm kAk1. In general, a bounded linear operator does not have eigenvalues. The closure of the finite rank operators in the operator norm forms the compact operators. A compact operator is well known to possess a discrete set of eigenvalues whose only limit point is zero, thus generalizing the eigenvalues of a matrix. The trace Tr of an operator A is finite (by whichp we mean the two sums on the right of (1) and (2) converge absolutely) if jAjD AA is a compact operator whose eigenvalues belong to the sequence space l1. Trace class operators, as the operators in the last sentence are called, form a two-sided ideal in the algebra of bounded operators, that is the sum of two trace class operators is a trace class operator and BA and AB are trace class operators if B is bounded and A is trace class. These observations and extrapolations (for instance, to ideals where the eigenvalues belong to l2 or lp,1 p<1) go back to Hilbert, Schmidt, von Neumann, and Schatten, and such ideals are the original symmetric operator spaces. Associated to each bounded operator B is the functional

'B : A ! Tr.BA/ on the trace class operators. In fact the bounded operators with operator norm are dual to the trace class operators under this mapping (where a trace class operator A is given the norm Tr.jAj/), like the Lebesgue L1-space is dual to the Lebesgue L1-space. There is much to this analogy. A successful theory of noncommutative integration has developed treating Tr like the integral, and the bounded operators and the trace have become the prototype for the theory of von Neumann operator algebras, colloquially known as noncommutative measure theory. Following the L1 and L1 analogy, the functional on the bounded operators of the form

'A : B ! Tr.BA/ (3) where A is trace class, is called a normal linear functional on the bounded operators, and it provides a “finite measure” subordinate to Tr. In 1966, toward the pinnacle of the semifinite standard theory of noncommutative integration but before Alain Connes carved his territory into the purely infinite theory, Jacques Dixmier published, in a two-page side note, an interesting construction. When A is compact and positive (A DjAj)Dixmierset 1 Xn 1 Tr .A/ :D ! .j , A/ (4) ! log.2 C n/ j D0 nD0 where the non-zero eigenvalues .n, A/, n 0, are ordered to be decreasing. Here ! is an extension of the usual limit on convergent sequences, that is, ! is a state on the algebra l1 that vanishes on c0. Dixmier’s construction can be applied to any compact Introduction 3 operator using linearity (every bounded operator is the complex linear sum of four positive operators). The functional Tr! is finite on the Lorentz ideal of compact operators A whose singular values (the non-zero eigenvalues of jAj ordered to be decreasing) satisfy 1 Xn .j , jAj/ D O.1/, n 0. (5) log.2 C n/ j D0

Thus, Tr!.A/ is ﬁnite for a trace class operator A. The functional Tr! is unitarily invariant since .n, jUAU j/ D .n, jAj/, n 0, for all isometries U of the Hilbert space. To make the functional Tr! linear, Dixmier asked that ! be dilation invariant, that is, invariant with respect to the dilation semigroup n : l1 ! l1, n 1, deﬁned by setting

n.a0, a1, :::/D .a„0, :::ƒ‚, a…0, a„1, :::ƒ‚, a…1, /. n times n times

Thus, Tr! is a trace (a unitarily invariant linear functional) on the Lorentz ideal of compact operators whose singular values satisfy (5), with the remarkable feature that it vanishes on the trace class operators,

Tr!.A/ D 0ifA is trace class.

The proof of this fact follows from the deﬁnition of the Dixmier trace in (4). If A is positive and trace class, then Xn .j , jAj/ D O.1/, n 0, j D0 and 1 Xn .j , jAj/ D o.1/, n 0. log.2 C n/ j D0

Hence, the Dixmier trace of A vanishes since an extended limit ! vanishes on c0. The trace Tr! cannot, therefore, be an extension of the canonical operator trace. The vanishing of Tr! where Tr is finite is the origin of the term singular trace. Dixmier’s construction opened a new field. Were Dixmier traces the only traces on this Lorentz ideal? Were there traces on ideals associated to other symmetric sequence spaces (and their semifinite von Neumann algebra equivalents)? What relationships are there between the canonical operator trace and singular traces? 4 Introduction

Answers to these questions are the theory of singular traces on symmetric operator spaces. Before we discuss some of the achievements in this theory, and the manner of interesting and inventive mathematics by which they are obtained—which is the content of two thirds of this book, let us tackle the remaining obligatory point. Is there anything to singular traces? What are they good for? What is their use?

What is a Singular Trace Good For? By their nature singular traces are very interesting. They cannot exist in ﬁnite linear algebra. They are spawned from the extra room in Hilbert space. Continuous singular traces such as Dixmier traces do not observe trace class perturbations at all, to them the trace class ideal is a zero-dimensional point. On the other hand, divergence for the canonical operator trace can be ﬁnite for a singular trace. Deep facts center around divergence of the harmonic sequence and Dixmier traces. This is more than a spurious coincidence. The connection between Dixmier traces and residues of -functions is part of the content we offer the reader. Dixmier’s construction results in linear functionals that are not normal, that is, they cannot be written in the form (3). Non-normal functionals lie outside the “measure class” of the canonical trace Tr, in particular, the functional on the bounded operators,

B ! Tr!.BA/,(6) with A in the Lorentz ideal that is the domain of the Dixmier trace, is continuous for the operator norm but it is not normal. In examples, the above functional is usually found to be an extension of a commutative or classical measure, by which we mean a Radon measure in the usual measure theory sense. This is a very interesting feature. For example, the canonical quantum mechanical pair .p, x/, represented as .id=dx, x/ on the Hilbert space L2.Œa, b/ of square integrable functions on an interval Œa, b of the real line, feature in such a situation with the operator A Djd=dxj1 linked to the 4 algebra L1.Œa, b/ of essentially bounded functions . Particularly, we know that, if f is an essentially bounded function on an interval Œa, b which acts on h 2 L2.Œa, b/ by h.x/ ! f.x/h.x/ almost everywhere, then ˇ ˇ Z ˇ ˇ1 b ˇ d ˇ 1 Tr! f.x/ˇ ˇ D f.x/dx. dx a

Thus, the linear functional on all bounded operators on L2.Œa, b/, ˇ ˇ ˇ ˇ1 ˇ d ˇ B ! Tr! Bˇ ˇ , dx

4 We impose Dirichlet boundary conditions so that the operator jd=dxj1 is compact. Introduction 5 is an extension of the Lebesgue integral from the (commutative) algebra of essentially bounded functions to the whole (noncommutative) algebra of bounded operators on L2.Œa, b/. This functional, unlike a normal linear functional, vanishes on all compact operators, which usually represent observables of localized quantum systems. This functional “sees” the classical system which is emergent in the weak closure of the localized quantum systems, but does not “see” perturbations due to localized quantum systems. This feature, which is a consequence of the Dixmier trace being a singular trace, represents a breakout from the “measure class” of the canonical trace Tr since the Dixmier trace vanishes on all finite rank projections, and the finite rank projections are the “atoms” of the measure theory associated to the canonical trace. These interests the reader can fairly say are speculative, philosophical, but not evi- dence of mathematical value. Alain Connes entered the fray in 1988, and set the world of singular traces on fire by observing the connection between singular traces and Mariusz Wodzicki’s noncommutative residue in differential geometry. In the same paper that Connes proved his trace theorem, he concretely linked Dixmier traces to Yang–Mills action functionals and posited the Dixmier trace as a replacement for taking the “classical limit”. Like other trace theorems, for instance Selberg’s Trace Formula, Connes’ Trace Theorem associates a spectral and essentially incalculable object to a calculable formula. If A is a classical pseudo-differential operator of order d on a d-dimensional closed manifold, then the extension of A, as an operator from square integrable functions to square integrable functions, is a compact operator that belongs to the domain of a Dixmier trace and

Tr!.A/ D ResW .A/ (7)

5 where ResW is Wodzicki’s noncommutative residue . The noncommutative residue is calculated from the principal symbol and has geometric meaning. This result was taken as a sign that the Dixmier trace, a singular trace, and it can only be a singular trace in the formula (7), should be the noncommutative residue for Connes’ new noncommutative differential geometry. This view has had great success and motivated much work to achieve similar geometric results for noncommutative tori, fractals, and foliations. Such successes have kindled hope of a common, operator-based, framework where the physical theories of quantum mechanics and general relativity may interact as perturbations and divergences of each other. As yet this hope is unfulﬁlled. Not counting ourselves physicists, there is no danger of it being fulﬁlled in this text. We do know singular traces however. We can now prove a conjecture on the link between the Gibbs state and the Dixmier trace. If D2 is a positive unbounded operator on a separable Hilbert space H , with compact resolvent

5 We always scale the usual formula for the noncommutative residue by the reciprocal of the dimension. 6 Introduction satisfying Weyl asymptotics like the Hodge–Laplacian on a closed manifold, that is,

2 2=d n n , n 0,

2 2 where n are the eigenvalues of D ordered to be an increasing sequence, then (we substitute .1 C D2/1=2 with the useful notation hDi) ˇ 1D2 d Tr.Be / Tr!.BhDi / D ! ı M Tr.eˇ 1D2 / for any bounded operator B on the Hilbert space H .HereM : L1.RC/ ! L1.RC/ denotes the continuous logarithmic means, Z 1 t ds .M x/.t/ :D x.s/ , x 2 L1.RC/, log.t/ 0 s and the composition ! ı M of an extended limit and the logarithmic mean also forms an extension of the usual limit on convergent at inﬁnity functions6. From work with Nigel Kalton we also know that 1 Xn 1 Tr .BhDid / D ! hBe , e i D ! ı M.fhBe , e ig1 / ! log.2 C n/ 1 C j j j j j j D0 j D0

1 2 2 2 where fej gj D0 is an eigenbasis of D ordered so that D ej D j ej , j 0. Here M denotes the discrete logarithmic means, P n 1 xj 1 Pj D0 1Cj 1 M.x/ :DfMn.x/gnD0 ,Mn.x/ :D n 1 , x DfxngnD0 2 l1, j D0 1Cj and the composition ! ı M forms an extension of the usual limit on convergent se- 2 1 quences. If D is thought of as the energy operator, then fej gj D0 is an eigenbasis of 1 increasing energy states and fhBej , ej igj D0 are the corresponding expectation values as the energy levels increase. So in the equalities ˇ 1D2 d 1 Tr.Be / Tr!.BhDi / D ! ı M.fhBej , ej ig D / D ! ı M j 0 Tr.eˇ 1D2 / we read that the high temperature limit of the observable B, the large quantum number limit, and the functional d B ! Tr!.BhDi /

6 We will be more precise about the extended limit ! in the text. Here it can be simply understood as an extension to the essentially bounded functions, or bounded sequences respectively, of the usual limit on convergent at inﬁnity functions, or the usual limit on convergent sequences respectively. Introduction 7 all agree. Moreover, these limits coincide in a unique “classical limit” with

ˇ 1D2 1 Tr.Be / B D lim ıM.fhBej , ej ig D / D lim ıM j 0 Tr.eˇ 1D2 /

d when B is such that the value Tr!.BhDi / is independent of the state ! (the notation B is the popular mathematical physics notation for the Dixmier trace in this situation). This mathematically unifies the “correspondence principle”, the “high temperature limit”, and singular traces. It also unifies uniqueness of classical limits with Connes’ notion of measurability. Thus, we find remarkable substance to Connes’ posit of singular traces as the replacement of taking the classical limit. This, we believe, is the ultimate value, the so what, of singular traces, and we are excited about the insights and applications coming from this direction.

How Are These Results Achieved?

Behind the single line equalities we have made in the previous page of text is extensive technical work. Now we must defer to that machinery and explain to the reader how such results, at least in the theory of singular traces on symmetric operator spaces, are obtained. This, in a nutshell, is the purpose of this book. The gap between von Neumann in 1932 and Lidskii in 1959 is due to the difﬁculty in proving Lidskii’s formula. For normal operators (of the form B C iC where B and C are Hermitian and BC D CB) Lidskii’s formula is trivial. The many years between the result for normal trace class operators and the result for any trace class operator can be summed up by one term: quasi-nilpotent. Nigel Kalton’s work, published in 1989, on the spectral character of the commutator subspace of the trace class operators, and later work with Ken Dykema on quasi- nilpotent compact operators, began our present understanding of commutator subspaces. Why are commutator subspaces and quasi-nilpotent operators vital and why do they form the cornerstone of trace theory on compact operators? For a two-sided ideal E of compact operators the commutator subspace is

Com.E/ :D Lin fŒA, B D AB BA : A 2 E, B is boundedg.

A linear functional on E, ' : E ! C, is a trace if and only if it vanishes on commutators. According to Calkin theory there is a sequence space E so that

E 1 A 2 ”f.n, jAj/gnD0 2 E. 8 Introduction

For example, in the case of the trace class operators (the domain of Tr), the sequence space is l1. In the case of the ideal of compact operators satisfying (5) (the domain of a Dixmier trace), the sequence space is the Sargent sequence space m1,1.When E 1 A 2 is normal, then f.j , A/gj D0 2 E,andA belongs to the commutator subspace Com.E/ if and only if the Cesàro mean of the eigenvalue sequence of A belongs back in the sequence space

1 Xn A 2 Com.E/ ” .j , A/ 2 E.(8) n C 1 j D0

This result was ﬁrst derived for l1 by Kalton and then extended to any E by Dykema, Figiel, Weiss and Wodzicki. How (8) is extended to non-normal operators is the deﬁning aspect of the Lidskii problem. The superdiagonal form of compact operators indicates that, to each compact operator A there is a compact normal operator NA with the same non-zero eigenvalues of A, with the same multiplicity, such that

Q :D A NA is quasi-nilpotent.

We recall a quasi-nilpotent operator has spectral radius

n 1=n lim kQ k1 D 0, n!1 however, unlike nilpotent matrices which have trace zero, if A 2 E there is no reason why Q 2 Com.E/. Kalton and Dykema found, essentially, necessary and sufﬁcient conditions for Q 2 Com.E/. It is not true that Q 2 Com.E/ generally, but it is true when E is a symmetrically normed ideal of compact operators. Having (8) for all A 2 E (for certain E as mentioned) provides fantastic results. From (8) it can be shown that

1 Xn 1 Xn A B 2 Com.E/ ” .j , A/ .j , B/ 2 E n C 1 n C 1 j D0 j D0 which is an entirely spectral condition for two operators to have the same trace for every trace on a two-sided ideal of compact operators E. For example, this shows that

1 E A diag.f.n, A/gnD0/ 2 Com. / P 1 1 where diag.f.n, A/gnD0/ D nD0 .n, A/en ˝ en and en ˝ en are the rank one 1 projections of any orthonormal basis fengnD0 of the Hilbert space. Thus, for any trace ' on E 1 '.A/ D ' ı diag f.n, A/gnD0 (9) Introduction 9

1 and ' is expressed as a linear functional on the eigenvalue sequence f.n, A/gnD0 2 E. Formula (9) we call the Lidskii formula for the trace '.WhenP E D l1, then Lidskii’s 1959 formula for Tr follows from the fact that Tr ı diag D . The Lidskii formula for the Dixmier trace of an arbitrary compact operator A with

1 Xn .j , jAj/ D O.1/, n 0, log.2 C n/ j D0 becomes 1 Xn 1 Tr .A/ D ! .j , A/ . ! log.2 C n/ j D0 nD0

We recall that .n, A/, n 0, are the non-zero eigenvalues which are enumerated so that j.n, A/j, n 0, are decreasing (or the zero sequence if A is quasi-nilpotent). Despite being exactly Dixmier’s definition when A DjAj the Lidskii formula for a Dixmier trace is not a trivial result, the time gap between von Neumann and Lidskii indicates the difficulty of proving such a formula. From Lidskii formulations of the Dixmier trace follows much of the theory about Dixmier traces. The -function residue formula and the heat kernel asymptotic formula for the Dixmier trace, so familiar to workers in noncommutative geometry and to workers in elliptic pseudo-differential operators, follow from Lidskii formulations. We explain how in this text. The value to the field of these other formulas is in providing calculation methods. In this book we add one new computation formula. Despite the Lidskii formula for the Dixmier trace, one is generally none the wiser about how to calculate the functional on the bounded operators

B ! Tr!.BA/ because the eigenvalue sequence .j , BA/, j 0, of a product is intractable. The following expectation value formula may be new to the reader, even to specialists in 1 the ﬁeld. If A DjAj and fengnD0 is an orthonormal basis of the Hilbert space such that Aen D .n, A/en, n 0, then 1 Xn 1 Tr .BA/ D ! hBe , e i.j , A/ ! log.2 C n/ j j j D0 nD0 for B bounded. The two formulas, Lidskii and expectation values, are vital to trace theory. As is seen in the applications, from them follow the classical limit characterization, part of the “so what”, of singular traces. From them follows Connes’ Trace Theorem (which we prove for a closed manifold and for any compactly based pseudo-differential operator 1 Rd 1 Rd Rd A : C . / ! Cc . / on ), and examples such as isospectral deformations. 10 Introduction

Are There Any Singular Traces Besides Dixmier Traces?

Thus far we have not mentioned any singular traces besides Dixmier traces. Our ﬁ- nal remarks concern this. We may have given the impression that Dixmier traces are everything about singular traces and their theory. They are and they are not. If E is a symmetrically normed ideal of compact operators, alongside the commutator subspace this text investigates the closed commutator space Com.E/, which is the closure of the commutator subspace in the norm of the symmetrically normed ideal. As mentioned in (8), the commutator subspace of E is characterized by

A 2 Com.E/ ” C..A// 2 E

1 where C denotes the Cesàro means and .A/ Df.n, A/gnD0 denotes an eigenvalue sequence of a compact operator A. The closed commutator space is characterized by 1 1 A 2 Com.E/ ” 1 C..A// ! 0inE (10) log m m m where m, m 1, denotes the dilation operation on sequences. The Calkin sequence space E associated to a symmetrically normed ideal E is itself symmetrically normed and kAkE Dk.jAj/kE . The convergence in (10) is in the norm of E. In the same way as the commutator subspace is the kernel of all traces on the ideal E, the closed commutator subspace is the kernel of all continuous traces. With (8) and (10) we can confirm or deny the existence of traces, or continuous traces, respectively, on the ideal E and whether the sets of the two are distinct, according to the invariance of E under the Cesàro operator C , or the vanishing of E under the condition in (10), respectively. Applied to the Lorentz ideal on which Dixmier traces are finite, we see that there exist singular traces that are not continuous, and hence there are singular traces that are not (linear combinations of) Dixmier traces. One might then raise the questions whether there are continuous singular traces that are not Dixmier traces, and if the answer is still in the negative, at least can continuous singular traces be approximated in the weak topology by Dixmier traces? The Lorentz ideal on which Dixmier traces are finite has additional structure; it is what we call a fully symmetric operator ideal. This means that its sequence space preserves Hardy–Littlewood(–Polya) submajorization. Let

Xn Xn A B be notation for .j , jAj/ .j , jBj/, n 0. j D0 j D0

A symmetrically normed ideal E is fully symmetric if B 2 E and A B implies A 2 E. A trace ' on E is fully symmetric if A DjAj is bounded, B DjBj2E, Introduction 11 and A B implies '.A/ '.B/.

The set of fully symmetric traces is weak closed. When the fully symmetric ideal E of compact operators admits a symmetric norm which is Fatou (meaning kAkE supn kAnkE if 0 An " A weakly, and all examples such as Schatten operator ideals, Lorentz operator ideals, Orlicz operator ideals, etc. have symmetric norms which are Fatou norms), there are the following mutually exclusive possibilities for traces on such an ideal: (a) E has no continuous trace (excepting the trivial zero functional);

(b) E has a (up to a constant factor) unique continuous trace, and that continuous trace extends the canonical operator trace;

(c) E has an infinite number of traces and the set of fully symmetric traces is strictly smaller than the set of continuous traces. Deriving these results using a functional analysis approach is one of the technical highlights of the book. What they imply for the Lorentz ideal on which Dixmier traces are finite, is that the set of fully symmetric traces is a weak closed strict subset of the set of continuous traces. The final remarkable piece of the puzzle is that the set of Dixmier traces is identical to the set of all normalized fully symmetric traces. Thus the answer is no, the Dixmier traces do not span the set of continuous traces on this ideal, nor is their span weak dense in the set of continuous traces. Now that the reader understands the position of Dixmier traces, we reach the final application of commutator subspaces that we would like to communicate. We hope to entice and interest even the specialist to read further into the book. We return to Alain Connes’ Trace Theorem. We show in the application section of this book that one can compute the Dixmier trace of any pseudo-differential operator A of order d on a closed manifold using the principal symbol (since A may not be classical, the principal symbol is an equivalence class). There are pseudo-differential operators A for which the value Tr!.A/ depends on the state !, that is, they are not measurable in Connes’ sense. On the other hand, when A is a classical pseudo-differential operator of order d, its extension belongs to the weak-l1 ideal of compact operators and

'.A/ D ResW .A/

1 1 for every trace ' with ' ı diag.f.1 C n/ gnD0/ D 1. Not just the Dixmier traces, nor the continuous traces, but every trace, as indicated by the uniqueness properties of the noncommutative residue shown by Wodzicki. The generalization of Connes’ Trace Theorem we provide also shows, as a corollary, that for every square integrable function f on a closed d-dimensional manifold X,the 12 Introduction

d=2 operator Mf .1 / belongs to the weak-l1 ideal of compact operators and Z Vol .Sd1/ '.M .1 /d=2/ D f.u/du. f d d.2/ X

Here is the Laplace–Beltrami operator on X, Mf is the closeable unbounded operator deﬁned by,

.Mf h/.u/ :D f .u/h.u/, h 2 L1.X/,

1 1 and ' is any trace with ' ı diag.f.1 C n/ gnD0/ D 1.

Structure of this Book The reader will find all the results discussed above in the book—and a few more. We have tried to provide a concise but complete reference to the theory of singular traces on symmetric operator spaces, and to not duplicate results readily available elsewhere. The Preliminaries (Part I) attempt to motivate later content without overwhelming the new or graduate reader. Part II concerns the general theory of continuous traces on symmetric operator spaces. It discusses the Calkin correspondence, which is the bijective correspondence between symmetric operator spaces and symmetric function spaces. Armed with this correspondence it shows the bijection between continuous traces on the operator space and symmetric functionals on the function space. The Lidskii formula for traces and commutator subspaces are then investigated. This part closes by showing that Dixmier traces (densely) provide all the fully symmetric traces on a fully symmetric ideal of compact operators. Part III concentrates on the consequences of the Lidskii formula for Dixmier traces on Lorentz ideals. In this part the -function residue and heat kernel formulas for Dixmier traces are shown. It finishes with an investigation of measurability in Connes’ sense, which is when an operator has the same value for every Dixmier trace. Part IV contains some applications of singular traces to noncommutative geometry. In considering the content of this part we chose against a catalog of applications from the literature. Given the foundation in Parts II and III we thought it more fitting to de- vote the space in the text, and the time and attention of the reader, to the new approach to noncommutative residues and integrals inspired by Kalton’s work on modulated operators. This part proves an extension of Connes’ Trace Theorem, and provides expectation value formulas for Dixmier traces. It shows that Connes’ noncommutative integral equates to the logarithmic mean of expectation values, and it finishes with the connection of the noncommutative integral to classical limits.

Part I Preliminary Material Chapter 1 What is a Singular Trace?

This chapter introduces singular traces on symmetric operator spaces assuming only a graduate level knowledge of topology and linear algebra. In what follows it is our intention to provide only those results, mostly without proof, that motivate the theory. Notation is also established. We refer to the monograph [188] for a graduate level introduction to functional analysis; for other references see the end notes to this chapter.

1.1 Compact Operators

Let H be a complex Hilbert space (an inner product vector space over the complex numbers that is complete in the associated vector norm). The inner product on H , denoted always hpx, yi, x, y 2 H , we take to be complex linear in the ﬁrst variable. Always kxk :D hx, xi, x 2 H , denotes the vector norm on H . A linear operator

A : H ! H is bounded if kAxkC kxk for all x 2 H for the same constant C>0. The smallest constant is the operator norm kAk1. We denote the set of bounded operators by L.H /. The bounded operators form the space of continuous linear operators of H to itself. In addition to the norm topology on the bounded operators given by the operator norm, the family of semi-norms fkAxk : x 2 H g forms a locally convex topology on the bounded operators called the strong operator topology. A vector space is separable if it has a countable basis. All abstract Hilbert spaces we consider will be separable and infinite dimensional; the finite dimensional theory is equivalent to linear algebra and can be obtained by restriction to a finite dimensional subspace. The Gram–Schmidt process familiar to linear algebra can be applied to infinite dimensional Hilbert spaces (in fact E. Schmidt applied the process explicitly to the Hilbert space l2 of square summable sequences, [210]). Lemma 1.1.1 ([210], [188, Theorem II.6]). A separable infinite dimensional Hilbert 1 spacePH admits a countable orthonormal basis fengnD0,i.e.hen, emiDımn and 1 x D nD0hx, enien for any x 2 H where the infinite sum is understood as the vector norm limit of partial sums and where ı denotes the Kronecker delta. 16 Chapter 1 What is a Singular Trace?

1 Z If fengnD0 is an orthonormal basis of H ,forﬁxedn, m 2 C, the linear operator

en ˝ em : H ! H deﬁned by

.en ˝ em/x :Dhx, eniem, x 2 H , is called a rank one projection. An operator is finite rank if it is a finite linear combination of rank one projections. We denote the finite rank operators by C00.H / and square N N matrices over the complex numbers by MN .C/. For a given finite rank operator 1 A there always exists some orthonormal basis fengnD0 so that A has a representation as a finite matrix,

C N A 2 00.H / ! ŒhAen, emin,mD0,:::,NA1 ,forsomeNA 2 .

Conversely, ﬁxing an orthonormal basis, any matrix algebra MN .C/ can be represented in the ﬁnite rank operators,

NX1 anmen ˝ em Œanmn,mD0,:::,N 1 2 MN .C/. n,mD0

1 Lemma 1.1.2. If fengnD0 is an orthonormal basis of the separable Hilbert space H and A 2 L.H / is a bounded operator then

X1 A D hAen, emien ˝ em n,mD0 where the inﬁnite sum is understood as a strong limit of partial sums. P 1 Proof.Sincex D mD0hx, emiem for any x 2 H ,then X1 X1 Ax D hAx, emiem D hAen, emihx, eniem. mD0 n,mD0

It follows that L.H / is the strong closure of the ﬁnite rank operators C00.H / and the theory of operators on inﬁnite dimensional Hilbert spaces is an extension of linear algebra. Results and terminology from linear algebra persist to operators, for example, the adjoint A : H ! H is the bounded operator with hAx, yiDhx, Ay i, x, y 2 H , and A is self-adjoint (or Hermitian) if A D A. A bounded operator A is positive (denoted A 0) if hAx, xi0 for every x 2 H , alternatively A D BB for some other bounded operator B. Section 1.1 Compact Operators 17

The extension of the spectral theory of matrices to operators is the most important tool in operator theory.

Deﬁnition 1.1.3. A complex number belongs to the resolvent .A/ of a bounded operator A 2 L.H / if A 1:H ! H is invertible, where 1:H ! H is the identity map. The resolvent is shown to be an open set [76, Theorem 1.4]. The complement, the spectrum, is therefore closed.

Deﬁnition 1.1.4. The spectrum .A/ of a bounded operator A 2 L.H / is the set C n .A/. Unlike matrices, for a bounded operator there is a distinction between being one- to-one and being invertible. The values 2 .A/ where A 1 is not one-to-one are termed eigenvalues since they imply the existence of an eigenvector x 2 H such that .A 1/x D 0.

Deﬁnition 1.1.5. The complex numbers 2 .A/ where A1 is not one-to-one are called the eigenvalues of A. A bounded operator need not have any eigenvalues or eigenvectors. The shift operator U1 : l2.Z/ ! l2.Z/ on the Hilbert space of double-sided square summable sequences, X1 1 1 2 U1 : fxngnD1 !fxnC1gnD1, jxnj < 1 nD1 is the standard example of a bounded operator without eigenvalues and eigenvectors, the details can found in introductory texts to functional analysis [188, 199]. A bounded operator is compact if it is the operator norm limit of a sequence of ﬁnite rank operators. We denote the set of compact operators by C0.H /. Compact operators have the advantage of possessing eigenvalues and eigenvectors.

Theorem 1.1.6 ([188, Theorem VI.15]). The spectrum of a compact operator consists of 0 and a countable set of eigenvalues whose only limit point, if it admits a limit point, is 0.

Deﬁnition 1.1.7. Let A 2 C0.H / be a compact operator and let be an eigenvalue of A. The eigenspace

E :Dfx 2 H : .A 1/x D 0g is a linear subspace of H . The dimension of the eigenspace E is called the geometric multiplicity of . 18 Chapter 1 What is a Singular Trace?

Deﬁnition 1.1.8. Let A 2 C0.H / be a compact operator and let be an eigenvalue of A. The set

N n R :Dfx 2 H : 9n 2 such that .A 1/ x D 0g is a linear subspace of H . The dimension of the linear space R is called the algebraic multiplicity of . The geometric multiplicity of an eigenvalue is less than or equal to the algebraic multiplicity.

Theorem 1.1.9 ([188, Theorem VI.15], [76, Theorem 2.21]). Let A 2 C0.H / be a compact operator and let be a non-zero eigenvalue of A. It follows that the algebraic multiplicity of is ﬁnite. The result is that, like matrices, the non-zero spectrum of a compact operator consists entirely of eigenvalues of ﬁnite algebraic multiplicity, and can be ordered by decreasing absolute value and associated to a sequence in c0.

Deﬁnition 1.1.10. An eigenvalue sequence

1 .A/ :Df.n, A/gnD0 2 c0 of a compact operator A 2 C0.H / is the sequence of non-zero eigenvalues .n, A/, n 0,ofA, repeated according to algebraic multiplicity, such that the absolute values j.n, A/j, n 0, are decreasing. If A has a ﬁnite number N of non-zero eigenvalues, we embedded the eigenvalue sequence in c0 as the sequence f.0, A/, :::, .N 1, A/,0,0,:::g2c00.IfA has no non-zero eigenvalues, the eigenvalue sequence of A is the zero sequence f0, 0, :::g. The appearance of eigenvalues in the text assumes that, unless stated explicitly, they are the (non-zero) eigenvalues of a compact operator and are ordered in an eigenvalue sequence. The ordering of eigenvalues in this way is not necessarily unique.

Example 1.1.11. The bounded sequences l1 can be embedded as bounded operators in any separable Hilbert space H .Leta 2 l1 act as an operator diag.a/ on H by, X1 1 diag.a/ :D anen ˝ en, a DfangnD0 2 l1 nD0

1 where en˝en are the rank one projections according to an orthonormalbasis fengnD0. Then diag.a/ : H ! H is a bounded linear operator with kdiag.a/k1 Dkak1 :D 1 supn2N janj. Finite sequences map to ﬁnite rank operators. If a DfangnD0 2 c0 then diag.a/ 2 C0.H / is compact and an eigenvalue sequence of diag.a/ is the non-zero values an ordered so that janj is decreasing (with zeros appended if a 2 c00). Section 1.1 Compact Operators 19

The operators in Example 1.1.11 are called diagonal operators. The spectral (or diagonalization) theorem of normal matrices can be extended to compact normal1 operators [188, Theorem VI.17, Theorem VII.1]. We write it in the form of eigenvalue sequences.

Theorem 1.1.12 (Spectral Theorem). Let A 2 C0.H / be a compact normal operator of rank N (where N D1is possible). Then the algebraic and geometric multiplicity N of each non-zero eigenvalue is equal and there exists an orthonormal basis fengnD0 of H ker.A/ consisting of eigenvectors of A such that

XN A D .n, A/en ˝ en, nD0 where the non-zero eigenvalues are listed according to algebraic multiplicity and en˝ N en are the rank one projections according to the eigenbasis fengnD0. If f 2 C..A// is a continuous function such that f.0/ D 0 then there is a compact operator

XN f.A/D f..n, A//en ˝ en nD0 on H such that .f.A// D f..A//. If A is a compact operator, by the spectral theorem there is a positive compact operator jApj, which is unique and which is termed the absolute value of A, such that jAj :D AA. This fact allows us to generalize singular values of matrices. C Deﬁnition 1.1.13. The singular value sequencep of a compact operator A 2 0.H / 1 is the eigenvalue sequence of jAj :D A A, denoted .A/ Df.n, A/gnD0 :D 1 f.n, jAj/gnD0. We will show properties of the singular value sequence when considering the generalized singular value function in Chapter 2, see also [222, Chapter 1]. Singular values generalize the commutative notion of decreasing rearrangement [149]. 1 C Example 1.1.14. If a DfangnD0 2 c0 then diag.a/ 2 0.H / is compact and the sequence of singular values is the decreasing rearrangement of the sequence a, denoted .a/ :D .diag.a// 2 c0. For any compact normal operator A, the sequence of singular values is the decreasing rearrangement of the eigenvalues of A,

.A/ Dj.A/j.

1 An operator A is normal if A D B C iC,whereB D B, C D C are Hermitian and BC D CB. 20 Chapter 1 What is a Singular Trace?

This is due to the spectral theorem,

.n, A/ D .n, jAj/ Dj.n, A/j, n 0.

The correspondence between singular values and eigenvalues breaks down for non- normal operators. A non-normal compact operator can have a zero eigenvalue sequence with non-zero singular values, as the following examples of quasi-nilpotent compact operators show.

Deﬁnition 1.1.15. A compact operator A 2 C0.H / is said to be (a) nilpotent if An D 0 for some n 2 N; (b) quasi-nilpotent if .A/ Df0g.

Example 1.1.16. An operator A : C2 ! C2 deﬁned by setting 01 A :D 00 is nilpotent. Indeed, A2 D 0. The eigenvalues are .A/ Df0, 0g yet the singular values are .A/ Df1, 0g. Evidently every nilpotent operator is quasi-nilpotent, but the converse is not true for operators which are not ﬁnite rank.

Example 1.1.17. The Volterra operator A : L2.0, 1/ ! L2.0, 1/ deﬁned by setting Z t .Ax/.t/ :D x.s/ds 0 is quasi-nilpotent but not nilpotent, see e.g. [136].

Proof. By induction Z t .t s/n .AnC1x/.t/ D x.s/ds, n 0. 0 nŠ The Neumann series converges for every ¤ 0 Z X1 t ..A 1/1x/.t/ D1 n.Anx/.t/ D1x.t/2 e.ts/=x.s/ds, nD0 0 hence .A 1/1 is bounded and .A/ Df0g. To show that A is not nilpotent it sufﬁces to observe that t n .An /.t/ D ¤ 0, 0

For every normal operator A (or for every quasi-nilpotent operator), the singular values of A dominate the modulus of an eigenvalue sequence of A. In general, singular values of non-normal operators need not dominate the modulus of eigenvalue sequences.

Example 1.1.18. There exists a matrix A 2 M2.C/ such that the inequality j.A/j .A/ fails.

Proof. Consider the following 2 2 matrix acting on C2, 11 A :D . 01

The eigenvalues of A are .0, A/ D 1and.1, A/ D 1. 10 11 11 jAj2 D AA D D 11 01 12 so jAj has the eigenvalues, listed in decreasing order, s p 3 C 5 .0, A/ D > j.0, A/j 2 and s p 3 5 .1, A/ D < j.1, A/j. 2

However, the sum (or product) of the singular values of a compact operator domi- nates the sum (or product) of the absolute value of an eigenvalue sequence. The following lemmas on compact operators on an inﬁnite dimensional separable Hilbert space H are due originally to H. Weyl [256].

Lemma 1.1.19. Let A 2 C0.H / be a compact operator. For every n 0, we have

Xn Xn j.k, A/j .k, A/. kD0 kD0

Lemma 1.1.20. Let A 2 C0.H / be a compact operator. For every n 0, we have

Yn Yn j.k, A/j .k, A/. kD0 kD0 22 Chapter 1 What is a Singular Trace?

A non-normal matrix M can be decomposed into the sum of a normal matrix (which is diagonalizable to a diagonal matrix that has the list of eigenvalues of M , with algebraic multiplicity, down the diagonal) and a nilpotent matrix. Since the matrix trace vanishes on nilpotent matrices, this reduces the trace of the non-normal matrix M to the trace of a normal matrix. The following result on the matrix trace rests on this fact.

N 1 C Theorem 1.1.21. Let M D ŒMnmn,mD0 2 MN . / be an N N matrix with complex terms. Then the matrix trace Tr of M (the sum of the diagonal terms) is equal to the sum of the eigenvalues of M repeated according to algebraic multiplicity:

NX1 NX1 Tr.M / D Mnn D .n, M/. nD0 nD0 Proof. Let M D N CR where R is nilpotent and N is normal such that an orthogonal matrix U exists with U NU D diagf.0, M/, :::, .N 1, M/g (the existence of U follows from the spectral theorem). Since Tr.R/ D 0, then

Tr.M / D Tr.N / D Tr.U NU/D Tr.diagf.0, M/, :::, .N 1, M/g/ since the matrix trace is invariant under orthogonal transformations.

The tactic of decomposition of a matrix into normal and nilpotent parts is of great advantage for traces in the inﬁnite dimensional theory also, where the decomposition of a compact operator is into the sum of a normal compact operator and a quasi-nilpotent compact operator. We will use the tactic later in the book. J. Ringrose extended the upper triangular form of matrices to compact operators [196], [197, Chapter 4]. The following decomposition was stated by T. West [255], in the context of Riesz operators, but for compact operators it follows directly from results on the upper triangular form of a compact operator in [196].

Theorem 1.1.22. Let A 2 C0.H / be a compact operator. There exists a compact normal operator N and a compact quasi-nilpotent operator Q such that A D N C Q and .A/ D .N /.

1.2 Calkin Correspondence

There is a remarkable correspondence between sequence spaces generated by singular values and two-sided ideals of compact operators, due to J. W. Calkin [26]. Two-sided ideals are tied to traces by equally remarkable results of N. Kalton and K. Dykema [81], and K. Dykema, T. Figiel, G. Weiss, M. Wodzicki [80], so that the Calkin correspondence indicating a plethora of two-sided ideals of compact operators brings with it a plethora of traces. This is a stark contrast to linear algebra where the matrix algebra has no proper ideals and admits only the matrix trace. Section 1.2 Calkin Correspondence 23

Deﬁnition 1.2.1. A linear subspace J of compact operators is a two-sided ideal if A 2 J and B 2 L.H / implies BA, AB 2 J .

Deﬁnition 1.2.2. A Calkin space J is a subspace of c0 such that a 2 J and .b/ .a/ implies b 2 J , where .a/ D .diag.a// is the decreasing rearrangement in Example 1.1.14.

Calkin spaces can also be viewed as follows. Let be a permutation of the set of non-negative integers. Suppose that i and j are non-negative integers such that .i/ D j . Consider the permutation matrix associated to : X1 … :D ….m, n/en ˝ em m,nD0 where ….i, j/D ….j , i/ D 1and….m, n/ D 0 for all other couples of non-negative integers m, n. If we identify the spaces l1 and c0 with the corresponding subsets of diagonal matrices in L.H / and C0.H / respectively, as done in Example 1.1.11, a Calkin space J is an arbitrary subset of c0 whichisanidealinl1 and which is rearrangement invariant in the sense that …J … J , for every permutation matrix ….Thatis,for any element a D diag.a/ 2 J ,wehave…a… D b 2 J where b D diag.b/ is given 1 by b Dfa.n/gnD0. The correspondence will be written by us in the following way. If J is a Calkin space then associate to it the subset J of compact operators

J :DfA 2 C0.H / : .A/ 2 J g.

Conversely, if J is a two-sided ideal associate to it the sequence space

J :Dfa 2 c0 :diag.a/ 2 J g.

For the proof of the following result we refer to Calkin’s original paper [26], and B. Simon’s book [222, Chapter 2]. A more general format of Calkin’s theorem and its proof is given in this book in Chapter 2.

Theorem 1.2.3 (Calkin correspondence). The correspondence J $ J is a bijection between Calkin spaces and two-sided ideals of compact operators.

On examining the form of the Calkin correspondence, we see that it is the statement that the diagonal of a two-sided ideal is a Calkin space and vice versa (the diagonal can be taken with respect to any orthonormal basis). Singular values

: JC ! JC 24 Chapter 1 What is a Singular Trace? provide the mapping from the positive operators in the operator ideal to the positive sequences in the Calkin space. The Calkin correspondence dictates our notation: we always denote ideals in commutative algebras (like l1) by straight letters (e.g. l1, c0, J ) and the corresponding noncommutative objects are denoted by curly letters (e.g. L.H /, C0.H /, J ). We consider the commutative sequence spaces embedded as the diagonal inside their noncommutative counterpart. The following sequence spaces and corresponding ideals are referenced repeatedly in the text.

Example 1.2.4. The sequence space of eventually vanishing sequences c00 c0 corresponds to the ideal of ﬁnite rank operators C00.H /. The sequence space c00 is the smallest permutation invariant sequence space, hence the ideal of ﬁnite rank operators forms the minimal two-sided ideal of compact operators. That is, if J is a two-sided ideal then C00.H / J .

Example 1.2.5. The lp sequence spaces, p 1, X1 1=p p lp :Dfa 2 c0 : kakp :D .n, a/ < 1g nD0 correspond to the Schatten–von Neumann ideals of compact operators X1 1=p p Lp :DfA 2 C0.H / : kAkp :D .n, A/ < 1g. nD0

Example 1.2.6. The weak-lp sequence spaces lp,1, p 1,

1=p lp,1 :Dfa 2 c0 : .n, a/ D O..1 C n/ /g correspond to the weak-lp ideals of compact operators

1=p Lp,1 :DfA 2 C0.H / : .n, A/ D O..1 C n/ /g.

Example 1.2.7. Let : RC ! RC be an increasing positive concave function. The Lorentz sequence spaces m and m are

1 Xn m :Dfa 2 c0 : kxkm :D sup .j , a/ < 1g n0 .n C 1/ j D0 and X1 m :Dfa 2 c0 : kxkm :D .n, a/. .n C 1/ .n// < 1g. nD0 Section 1.2 Calkin Correspondence 25

The corresponding Lorentz ideals of compact operators are

Xn M C 1 :DfA 2 0.H / : kAkM :D sup .j , A/ < 1g n0 .n C 1/ j D0 and X1 M C :DfA 2 0.H / : kAkM :D .n, A/. .n C 1/ .n// < 1g. nD0

L M 11=p Since l1 m then 1 .If p.t/ :D t , p>1,thenm p D lp,1 so the weak-lp ideals are Lorentz ideals for p>1.

Let :D log.1 C t/, t>0. Deﬁne the Lorentz sequence space

1 Xn m1,1 :Dfa 2 c0 : kakm1,1 :D sup .j , a/ < 1g. n0 log.2 C n/ j D0

Lemma 1.2.8. The inclusions l1 l1,1 m1,1 are strict.

Proof. To see that l1 l1,1 it is sufﬁcient to show that a D .a/ 2 l1 belongs to l1,1. Fix an arbitrary element a D .a/ 2 l1 and observe that

Xn .n C 1/ an ak kak1, n 0. kD0

1 Therefore, an D O..1 C n/ /, n 0, and a 2 l1,1. The inclusion is strict since the 1 1 sequence f.1 C n/ gnD0 does not belong to l1 and obviously belongs to l1,1.Tosee 1 1 that l1,1 m1,1 it is sufﬁcientP to check that f.1 C n/ gnD0 belongs to m1,1.The N 1 veriﬁcation is immediate, since nD0.1 C n/ 2log.2 C N/for every N 0. To see that the latter inclusion is strict, consider the following sequence

k k2 .kC1/21 a0 D 1, an :D ,2 n 2 , k 0. 2.kC1/21 2k2

.kC1/21 We claim that a 2 m1,1. Indeed, for every natural N such that N D 2 for some k 0, we have

XN XN k.k C 1/ 2 .n, a/ D a D 1 C 1 C 2 C 3 CCk D 1 C log.2.kC1/ / n 2 nD0 nD0 26 Chapter 1 What is a Singular Trace?

P N and this, together with the observation that the mapping N ! nD0 .n, a/, N 1 is piecewise linear concave, yields the required estimate

XN .n, a/ 2log.N C 2/, N 1. nD0

At the same time a … l1,1. To see this, it is sufﬁcient to verify that for every C>1 1 1 1 the inequality fangnD0 C f.nC1/ gnD0 fails. Take an arbitrary C>1 and choose 2 .k0C1/ 1 k0 1sothatC k0.TakeN D 2 . By construction we have aN D k0 , which is strictly greater than k0 C . .k C /2 k2 .k C1/21 N 2 0 1 12 0 2 0 Example 1.2.9. In Alain Connes’ noncommutative geometry the Lorentz ideal of compact operators Xn M C 1 1,1 :DfA 2 0.H / : kAkM1,1 :D sup .j , A/ < 1g n0 log.2 C n/ j D0 associated to the sequence space m1,1 has been labeled with similar notation to the weak-l1 ideal L1,1. Since the inclusions of ideals

L1 L1,1 M1,1 is strict by the previous lemma and the Calkin correspondence, we seek to avoid confu- sion by using the notation M1,1. See the end notes for historical origins of the ideals L1,1 and M1,1. A bounded operator U : H ! H is a unitary if U U D UU D 1where1 is the identity map 1 : H ! H . As in linear algebra, unitaries are equivalent to 1 transformations of orthonormal bases, that is, fUengnD0 is an orthonormal basis if and 1 only if fengnD0 is, and the unitaries form the group of isometries of H . A trace on a two-sided ideal of compact operators is deﬁned in analogy to the fact that the matrix trace is invariant under orthogonal transformations.

Deﬁnition 1.2.10. Atrace' on a two-sided ideal of compact operators J is a unitarily invariant linear functional, that is ' : J ! C and '.UAU / D '.A/ for all A 2 J and all unitaries U 2 L.H /. Unitary invariance coincides with the notion that a trace vanishes on commutators.

Lemma 1.2.11. A linear functional ' : J ! C is a trace if and only if

'.AB/ D '.BA/, A 2 J , B 2 L.H /

(alternatively '.ŒA, B/ D 0 where ŒA, B :D AB BA). Section 1.2 Calkin Correspondence 27

Proof. If the condition holds, '..UA/U / D '.U UA/ D '.A/ for every unitary U 2 L.H /. Hence ' is unitarily invariant. For the “only if” assertion, notice that '.UA/ D '.U .UA/U / D '.AU/ and '.ŒA, U/ D 0 for every unitary U 2 L.H /. Every B 2 LP.H / can be written as a linear combinationP of four unitaries [188, 4 4 p. 209], B DP iD1 ˛i Ui where Ui Ui D I .ThenŒA, B D iD1 ˛i ŒA, Ui .Hence 4 '.ŒA, B/ D iD1 ˛i '.ŒA, Ui / D 0. Singular values are unitarily invariant, .UAU / D .A/ for all unitaries U 2 L.H / and compact operators A 2 C0.H /. Thus, using the Calkin correspondence

: JC ! JC, we can associate to every linear functional 'O : J ! C on the Calkin space J of a two-sided ideal of compact operators J the unitarily invariant functional

A !O'..A//, A 0. (1.1)

If, in addition, 'O vanishes on the subset

Z.J / :Df.A C B/ .A/ .B/ :0 A, B 2 J g then (1.1) is linear and, by linear extension2, deﬁnes a trace on J . Nigel Kalton called the set Z.J / the center of the Calkin space J . By remarkable results of Kalton and Dykema, Figiel, Weiss and Wodzicki, every trace on a two-sided ideal arises this way. Traces of positive operators are completely determined by their action on the diagonal. The Calkin correspondence provides the inverse mechanism, in that, if ' : J ! C is a trace then Dykema, Figiel, Weiss and Wodzicki’s results imply that

'.A/ D '.diag..A///,0 A 2 J .

The functional 'O :D ' ı diag is linear on the Calkin space J ,vanishesonZ.J /, and (1.1) is satisﬁed. Therefore,

2 Like matrices, every bounded operator can be decomposed into self-adjoint components

A D A1 C iA2 where A1 D 1=2.ACA / and A2 Di=2.AA /. Every self-adjoint operator A D A decomposes into positive components A D AC A

where AC D 1=2.A CjAj/ and A D1=2.A jAj/. Thus, any bounded operator is the complex linear combination of four positive operators. When we talk of “linear extension” we meanP that a quan- 4 tity ' deﬁnedP on positive operators can be extended to bounded operators by '.A/ D iD1 ˛i '.Ai / 4 if A D iD1 ˛i Ai , Ai 0. 28 Chapter 1 What is a Singular Trace?

the study of traces (linear functionals on a two-sided ideal that vanish on commutators) is equivalent to the study of linear functionals on the Calkin space that vanish on the center.

The reduction to sequence spaces is why we can answer so many questions about existence, uniqueness, and other properties of traces. It is not the scope of this text to develop the results of Kalton and Dykema, Figiel, Weiss and Wodzicki for general two-sided ideals. We concentrate on the development of the theory of traces for symmetric ideals of compact operators and their generalization to symmetric operator spaces. A symmetric ideal is a normed two-sided ideal of compact operators which is closed in the norm, and the norm is continuous for the two-sided action of the bounded operators.

Deﬁnition 1.2.12. A two-sided ideal E of compact operators is called a symmetric ideal if it is a Banach space for a norm kkE with the following “symmetric” property:

kBAC kE kBk1kAkE kC k1, A 2 E, B, C 2 L.H /.

Symmetric ideals of compact operators are also referred to as Banach ideals, and as symmetrically normed ideals in the terminology of I. Gohberg and M. Krein [98]. Symmetric ideals of compact operators have been extensively studied and feature in physical applications, Gohberg and Krein [98], M. Sh. Birman and M. Solomyak [12] and Simon [222]. What makes symmetric ideals of interest is that, as a Banach space, they have a Banach dual E. In this case it is natural to consider the continuous traces, where ' is a trace and ' 2 E . The theory of traces on symmetric ideals of compact operators will be our concern in Part II. Results there include, for symmetric ideals, the commutator subspace results of Kalton and Dykema, Figiel, Weiss and Wodzicki, and the result, only recently obtained, that the Calkin correspondence is an isometric correspondence between a symmetric ideal and a symmetric sequence space.

1.3 Examples of Traces

The matrix trace is a trace on the two-sided ideal of finite rank operators. We illustrate (1.1) with two other examples. Though by no means exhausting the construction of traces they are, because of their pervasiveness, the primary examples for this book. The first example is an extension of the matrix trace, written in abstract form by J. von Neumann [173] and associated to the symmetric ideal of trace class operators by R. Schatten and von Neumann [205, 207, 208]. Recall that every two-sided ideal contains the finite rank operators, so the question is raised whether every trace is an extension of the matrix trace. The second example is the Dixmier trace, due to J. Dixmier [62], which is a trace whose restriction to the finite rank operators van- Section 1.3 Examples of Traces 29 ishes; answering in the negative the question of whether every trace on the compact operators is just an extension of the matrix trace.

1.3.1 The Canonical Trace

Let 0 A 2 L1,whereL1 is the Schatten–von Neumann ideal of compact operators in Example 1.2.5 for p D 1. Then the sequence of singular values is summable, .A/ 2 l1. The canonical linear functional on l1 is the sum, so deﬁne X1 Xn Tr.A/ :D .n, A/ D lim .j , A/, A 0. (1.2) n!1 nD0 j D0

The functional Tr is linear if the sum vanishes on Z.l1/. We can avoid a direct computation by using two facts: (i) our knowledge from the spectral theorem that every 1 positive compact operator has an orthonormal basis ffngnD0 of H ker.A/ with Af n D .n, A/fn,then X1 Tr.A/ D hAf n, fni, A 0; nD0

1 and (ii) under summation, for any orthonormal basis fengnD0 of H , X1 X1 Tr.A/ D hAf n, fniD hAen, eni, A 0. nD0 nD0 By expressing Tr using expectation values it is immediate that Tr is linear and, of course, its linear extension to all A 2 L1 coincides with the abstract formulation of the trace as the “sum of the diagonal” X1 Tr.A/ D hAen, eni, A 2 L1. nD0

In this form Tr is an extension of the matrix trace. An operator A 2 L1 is called a trace class operator.

1.3.2 The Dixmier Trace

Let 0 A 2 M1,1 where M1,1 is the Lorentz ideal of compact operators in Exam- ple 1.2.9. Generally, the partial sums

Xn .j , A/ D O.log.2 C n//, A 0, n 0 j D0 30 Chapter 1 What is a Singular Trace? diverge, so it is not possible to use the sum, and the renormalization

1 Xn .j , A/ D O.1/, A 0, (1.3) log.2 C n/ j D0 is not convergent. To obtain a scalar value, the renormalized sequence (1.3) can be combined with an element ! 2 l1 from the dual space of the bounded sequences, 1 Xn 1 Tr .A/ :D ! .j , A/ , A 0. (1.4) ! log.2 C n/ j D0 nD0

The positivity and linearity of Tr! requires constraints on the element !. In the course of this book, the discussion and the choice of particular conditions imposed on ! will take a significant role due to the many formulas associated to the Dixmier trace. It is clear that if we would like Tr! to be positive, we have to assume that ! 0. The difficulty in the formulation (1.4) (and, generally, in the formulation (1.1)) is its linearity. What is interesting is that the linearity of Tr! implies that ! is a singular element of l1. We recall a few definitions. Denote by .l1/n and .l1/s the normal and singular parts of l1 [122]. The normal part is equivalent to l1, .l1/n Š l1 and the singular part .l1/s is such that

l1 D .l1/n ˚ .l1/s .

This formulation, and terminology, is the discrete analogy of the decomposition of Borel measures on Œ0, 1/ into absolutely continuous and singular parts. The following theorem shows, in particular, that if (1.4) is linear, then 0 ! 2 l1 must be singular. Conversely, if 0 ! 2 l1 is singular and of the particular form where it is invariant under the dilation operator n : l1 ! l1, n 1,

n.a0, a1, :::/:D .a„0, :::ƒ‚, a…0, a„1, :::ƒ‚, a…1, /, n times n times then (1.4) deﬁnes a linear functional.

Theorem 1.3.1. Let 0 ! 2 l1 be a positive linear functional on the algebra l1.

(a) If Tr! is additive on the positive cone of M1,1,then! is singular.

3 (b) If ! is singular and if ! ı n D !, n 2 N, then Tr! is additive on the positive cone of M1,1 (and hence extends to a positive trace on M1,1/.

3 It is sufﬁcient to require ! ı n D ! for some n 2. Section 1.3 Examples of Traces 31

Proof. The ﬁrst part is rather simple to prove, while the second part is less obvious. (a) Fix m 2 N.Letp, q 2 L.H / be orthogonal projections such that Tr.p/ D Tr.q/ D m.Letm denote the sequence f1, :::,1,0,:::g with m consecutive 1’s. We have .p/ D .q/ D m and .p C q/ D 2m. It follows from (1.4) that n o minfn C 1, 2mg Tr!.p C q/ D ! , log.2 C n/ n0 n o minfn C 1, mg Tr!.p/ D Tr!.q/ D ! . log.2 C n/ n0

Since Tr!.p C q/ D Tr!.p/ C Tr!.q/, it follows that n o 2minfn C 1, mgminfn C 1, 2mg ! D 0. log.2 C n/ n0

Since ! is a positive functional, it follows that !.m/ D 0. Thus, ! vanishes on an arbitrary ﬁnite sequence.

(b) Let A, B 2 M1,1 be positive operators. We need the following fundamental inequalities for singular values. These inequalities are proved in Section 3.3 (see Theorem 3.3.3 and Theorem 3.3.4) and play an indispensable role in this book. For every n 0, we have

Xn Xn 2XnC1 .k, A C B/ .k, A/ C .k, B/ .k, A C B/. kD0 kD0 kD0

The ﬁrst inequality immediately implies that

Tr!.A C B/ Tr!.A/ C Tr!.B/.

The second inequality implies that

n 2 C1 o 1 Xn Tr!.A/ C Tr!.B/ ! .k, A C B/ . log.2 C n/ n0 kD0

Since ! D ! ı 2, it follows that

n n 2Œ C1 o 1 X2 Tr!.A/ C Tr!.B/ ! n .k, A C B/ . log.2 C Œ / n0 2 kD0 32 Chapter 1 What is a Singular Trace?

Since ! is singular, it follows from the inequality

2Œ n C1 1 X2 1 Xn .k, A C B/ D .k, A C B/ C o.1/ log.2 C Œ n / log.2 C n/ 2 kD0 kD0 that

Tr!.A/ C Tr!.B/ Tr!.A C B/.

Thus, Tr! is additive on the positive cone of M1,1. Hence, Tr! extends to a linear functional on M1,1. The latter functional is unitarily invariant and is, therefore, a trace. 1 The singular elements 0 ! 2 .l1/s which are states (! f1gnD0 D 1) extend the ordinary limit: 1 1 ! fang D D lim an, fang D 2 c, n 0 n!1 n 0 from the space c of all convergent sequences to l1. The formulation (1.4) and the statement of Lemma 1.3.1 (b) for dilation invariant singular states are credited to Jacques 4 Dixmier . The resulting trace Tr! on M1,1 is named a Dixmier trace. Notice that 1 1 1 Xn 1 1 Tr diag D ! D 1 ! n C 1 log.2 C n/ j C 1 nD0 j D0 nD0 so that Dixmier traces are not trivial. Notice also that, for a trace class operator 0 A 2 L1 M1,1, 1 Xn .j , A/ D o.1/ log.2 C n/ j D0 as Xn .j , A/ D O.1/ j D0 and then

Tr!.A/ D 0, 0 A 2 L1,

since any 0 ! 2 .l1/s vanishes on o.1/-sequences. Thus, Dixmier traces are not extensions of the canonical trace Tr, and, in fact, vanish where Tr is ﬁnite. Such traces are examples of singular traces.

4 The existence of extended limits ! satisfying ! ı n D !, n 2 N, is established in Chapter 6. Section 1.3 Examples of Traces 33

Definition 1.3.2. Atrace' : J ! C on a two-sided ideal J is a singular trace if it vanishes on the subspace C00.H / of finite rank operators. Singular traces, since they vanish on all embeddings of matrices as finite rank operators, have no analog in linear algebra. In Part II we will see that all continuous traces ' 2 E on a symmetric ideal E of compact operators decompose either into a normal trace 'n (which is the extension Tr of the matrix trace when E D L1, or otherwise is trivial) and a singular trace 's,

' D 'n C 's .

When restricted to the minimal ideal of ﬁnite rank operators all continuous traces thus collapse either into the matrix trace or zero, resulting in the unique situation of linear algebra. There is an undiscovered country of traces in Hilbert space theory beyond the matrix trace.

There are many questions concerning Dixmier traces without needing to search for more exotic constructions of singular traces. The popularity of Dixmier traces in Alain Connes’ noncommutative geometry means that we spend Part III on their theory and Part IV on applications of singular trace theory to noncommutative geometry.

1.3.3 Lidskii Formulation of Traces Theorem 1.1.21 stated the well-known result that the matrix trace in linear algebra can be written equivalently as the sum of the eigenvalues of the matrix. Is the same true for the extension of the matrix trace to trace class operators? Can the Dixmier trace be written as a formula involving eigenvalues? 1 V. B. Lidskii showed in 1959 that, if .A/ Df.n, A/gnD0 is an eigenvalue sequence of a trace class operator A 2 L1,then X1 Tr.A/ D .n, A/, A 2 L1. nD0

This result does not follow trivially from (1.2), so much so that the formula is called the Lidskii formula. For a general trace ' on a two-sided ideal J , the Lidksii formula for ' is the demon- stration that the linear extension of

'.A/ D ' ı diag..A//, A 0 is, provided the eigenvalue sequence .A/ belongs to the Calkin space J , equivalent to '.A/ D ' ı diag..A//, A 2 J . 34 Chapter 1 What is a Singular Trace?

The Lidskii result is not trivial because, as we recall from the examples of quasi- nilpotent operators in Section 1.1, eigenvalue sequences are not a linear extension of the singular value sequence

: JC ! JC.

Kalton and Dykema [81], in their solution to the Lidskii problem, gave an example of a two-sided ideal J with A 2 J such that .A/ … J where J is the associated Calkin space. That is, for any consistent choice of ordering of eigenvalue sequences, in general, : J 6! J

(and also this indicates why the Calkin correspondence is based on singular values and not eigenvalues). Kalton and Dykema identiﬁed a class of ideals J for which

: J ! J does hold, and that, in the case of this class, the Lidskii formulation holds for all traces on J . In Part II, following the technique of Kalton and Dykema, it is shown that all traces on symmetric ideals of compact operators satisfy the Lidskii formula. In Part III we will look at speciﬁc Lidskii formulas for Dixmier traces. The pivotal trace theorem in noncommutative geometry in Part IV rests on the Lidskii formula and an estimate for eigenvalues of integral operators ‘modulated’ by the Laplacian.

1.4 Notes

Compact Operators For a history of the developmentof functionalanalysis we refer the reader to J. Dieudonne [59], or the encyclopedic work of A. Pietsch [182]. An approachable introduction to the modern treatment of functional analysis is [188] by M. Reed and B. Simon. Innumerable works on C -algebras and von Neumann algebras include the treatment of operators on Hilbert spaces, e.g. [63,243]. Reference works are the tomes of N. Dunford and J. Schwartz [78,79]. J. Rether- ford has written an introductionto functional analysis speciﬁcally aimedat an elementaryproof of the Lidskii trace theorem [190].

Spectral Theory The spectral theorem is probably the most important result in operator theory. Its milestones include D. Hilbert in 1906 [114], and F. Riesz in 1918 [193], who gave a complete treatment of what we now know as compact operators. Amazingly, Hilbert conceived of the continuous spectrum in 1906, and an associated spectral theorem in this case. Von Neumann in 1929 [170] and von Neumann [173], and M. Stone [226] in 1932 derived the modern version of the spectral theorem of bounded and unbounded operators and the abstract notion of a Hilbert space. L. Steen provides highlights in the history of the spectral theory in [224]. Section 1.4 Notes 35

H. R. Dowson provides a recommended exposition of the spectral theorem of Riesz operators and its historical development in Chapter 1 of [76].

Decomposition of Compact Operators The concept of quasi-nilpotent operator originated from the theory of integral equations [249]. The decomposition of a compact operator into a normal compact operator (with the same eigenvalues with algebraic multiplicity) and a quasi-nilpotent compact operator follows from Theorem 2, Theorem 6 and Theorem 7 of J. Ringrose’s 1962 paper [196]. T. West gave the explicit statement in the context of Riesz operators in Section 7 of the 1966 paper [255]. Every compact operator on a separable Hilbert space is a Riesz operator. An exposition features in Chapter 3 of the monograph [76] by Dowson. For the decomposition theorem see also [77, Theorem 8] in the context of spectral operators, but note that not all compact operators are spectral operators in the sense of Dunford.

Calkin correspondence The fundamental 1943 paper of J. W. Calkin [26], was motivated by the “observation that the ring L.H / of bounded everywhere deﬁned operators in Hilbert space contains nontrivial two-sided ideals. This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates L.H / sharply from the ring of all linear operators over a unitary space with ﬁnite dimension number.” The observation led Calkin to study and describe these ideals in terms of (commutative) ideals of sequence spaces and also led to the discovery of the Calkin algebra. Let H be a separable Hilbert space and let C0.H / be the ideal of all compact operators on H . The ideal C0.H / is norm closed in L.H /. The quotient C -algebra C D L.H /=C0.H / is called the Calkin algebra. It is a noncommutative analog of the quotient algebra l1=c0. The sequence spaces lp, lp,1, p 1, and m1,1, and their corresponding ideals, Lp, Lp,1 and M1,1, are extensively studied. M. Sh. Birman, M. Z. Solomjak and others have derived a formidable collection of spectral and norm estimates [12,14]. See also B. Simon’s book [222]. The notations of L1,1 and l1,1 came from the theory of weak-l1 spaces, see e.g. [11]. The notations for symmetrically normed ideals used by Gohberg and Krein in the books [98,99] are completely different, and inconsistent with the currently accepted notations of corresponding commutative objects like Lorentz spaces, see e.g. [10, 139]. We also refer to the paper of Pietsch [183], where he discussed the origin of the symmetrically normed ideal M1,1 and commutative space m1,1. The latter object was introduced by Sargent [203].

Traces Traces on operators first appeared in I. Fredholm’s 1903 work [93] on integral operators. Solving an integral equation for continuous functions in a manner similar to solving linear equations, Fredholm developed an infinite dimensional version of the determinant (see also H. von Koch [135]). The linear co-efficient in the Fredholm series representing the infinite dimensional characteristic equation, that is, the trace, took the form Z b K.x, x/dx a 2 for an integral operator SK with symmetric kernel K in C.Œa, b /. 36 Chapter 1 What is a Singular Trace?

D. Hilbert in 1904 [113] and E. Schmidt in 1907 [209] developeda spectral theorem for such integral operators, indicating the existence of an orthonormal system of continuous eigenvec- 1 1 tors fungnD0 and eigenvalues fngnD0 (the terms were introduced by Hilbert in [113]) with X1 K.x, y/ D nun.x/un.y/, nD0 or in modern terms the spectral theorem X1 SK D nun ˝ un. nD0 Hilbert’s work in 1906 on orthonormal systems [114], considered by Dieudonne [59, p. 110] as the paper establishingP functional analysis as a new field in itself, allowed it to be shown 1 1 when K.x, x/ > 0and nD0hSK en, eni < 1 for a complete orthonormal system feng0D1 Z b X1 X1 K.x, x/dx D n D hSKen, eni, a nD0 nD0 see J. Mercer [162]. Thus, the linear co-efficient in the Fredholm series becomes recognizable to modern eyes as the canonical trace of a positive integral operator with continuous kernel. While it was clear this linear coefficient was an extension of the matrix trace (Koch [135]), it was not named explicitly by these authors as a trace. The general theory of bounded and compact operators developed over the period 1906– 1932, and, according to Pietsch [182, p. 137], J. von Neumann in 1932 [173] first explicitly wrote the trace (or ‘Spur’) X1 Tr.S/ :D hSen, eni nD0 1 of a positive trace class operator S in an abstract Hilbert space with orthonormal basis fengnD0, and explicitly showed independence of the value from the underlying orthonormal basis. In fact, von Neumann wrote it for all positive operators, allowing the value 1, and foreshadowed the notion of a semifinite trace. From the spectral theorem, by then established for all self- adjoint compact operators, it was clear that the trace of a positive trace class operator was equal to the sum of its eigenvalues. The same statement for every trace class operator would have to wait until V. B. Lidskii in 1959 [148], though there are indications that A. Grothendieck knew the result in 1955 [182, p. 404]. Schatten and von Neumann in 1946 [205,207,208] established the ideal of trace class operators and its duality with the compact operators, and pre-duality with the bounded operators. Inspired by the trace class operators (of Schatten and von Neumann), the study of traces on nuclear operators on Banach spaces (that are not Hilbert spaces) was initiated by Ruston and Grothendieck, see [180]. This theory is more complicated, the matrix trace may not extend to the class of all nuclear operators [180, p. 213]. From the origin of functional analysis there has been a fundamentalinterplay between traces and determinants of linear operators, see [97] and [180].

Construction of Traces J. Dixmier’s construction [62] will feature heavily in this book. We mention other constructions. Section 1.4 Notes 37

Unknowingly of the paper by J. Dixmier [62], A. Pietsch approached the problem of existence of singular traces on operator ideals (in the framework of his general theory of operator ideals) from the axiomatic point of view. The main open problem suggested by Pietsch was “Whether there exist quasi-Banach operator ideals that support different continuous traces”? A solution to this problem was given by Nigel Kalton and can be found in his paper [123]. This work, in fact, precipitated Kalton’s work on commutators and although in this book we practically do not touch upon the connections with the Banach space theory which prompted Pietsch and Kalton to introduce and study singular traces, these connections appear to be rich and nontrivial. Pietsch’s recent paper [184] details a complete construction of traces on the ideal M1,1 and its Banach space equivalent. It also notes the unpublished contributions of T. Figiel to the theory of constructing traces on ideals of compact operators. In the paper [245], J. Varga characterized those compact operators A such that certain ideals associated with A admit nontrivial traces. For an exposition of Varga’s approach and the extension and generalization of the approach to general semiﬁnite von Neumann algebras, we refer to [67] and [103]. In the paper [259], M. Wodzicki introduced a concept for a Dixmier trace on an arbitrary operator ideal.

Lidskii Formula Lidskii in 1959 [148] used determinants to prove that the trace of every trace class operator is given by the sum of its eigenvalues. As mentioned above, Retherford’s text [190] is devoted to an elementary proof of Lidskii’s trace theorem. Also as mentioned, the result is not emulated in general Banach spaces. Determinant free proofs of Lidskii’s theorem that are based on the Ringrose upper diagonal form, such of those of Erdos [86] and Power [187], are those that lend themselves to extension to other operator ideals in the Hilbert space theory. Kalton and Dykema in 1998 [81] essentially solved the question of whether every trace on a two-sided ideal J satisﬁes the Lidskii formula. The technique of Kalton and Dykema will be the basis of our approach in Chapter 5, so we reserve further remarks until then. Chapter 2 Preliminaries on Symmetric Operator Spaces

This chapter provides prerequisites in the study of singular traces on symmetric operator spaces. The previous chapter was aimed at motivation, and it made no attempt to introduce the machinery of the general theory beyond the basics of functional analysis. We introduce von Neumann algebras and semiﬁnite traces, the singular value function, and symmetric operator spaces, which are the central components of the general theory. The background and context for these components is extensive. We reference here only those parts of the theory required for later sections, or as motivation for later material; the theory is comprehensively treated elsewhere in, for example, [63, 185, 243] (see also [68]).

2.1 Von Neumann Algebras

In this section we introduce von Neumann algebras (with separable pre-dual). The theory of von Neumann algebras extends the measure theory of function algebras to general operator algebras. The synthesis of operator theory and measure theory provides a framework that incorporates both linear algebra and classical function spaces. Let H be a separable Hilbert space. We consider three basic topologies on L.H /. The norm (or uniform) topology on L.H / is given by the operator norm kk1 as mentioned in Chapter 1. The strong operator topology, also mentioned, is the topology of pointwise convergence on L.H /: a basis of the neighborhoods of an element A 2 L.H / in this topology is given by the sets

n L U.A, fj gj D0/ :DfB 2 .H / : k.B A/j k, 8j g, >0, n 0.

The weak operator topology on L.H / is the topology in which the basis of the neighborhoods of an element A 2 L.H / is given by the sets

n n L V.A, fj gj D0, fj gj D0/ :DfB 2 .H /: jh.BA/j , j ij , 8j g, >0, n 0.

n n Here fj gj D0 and fj gj D0 are arbitrary sequences of elements of H . Let M be a subset of L.H /. The commutant M0 is the set

M0 :DfB 2 L.H / : AB D BA, 8A 2 Mg. Section 2.1 Von Neumann Algebras 39

The bicommutant M00 of M is the commutant of M0. The intersection of M and its commutant M0 is called the center of M. The famous von Neumann bicommutant theorem [171], [19, §2.4], yields that a unital self-adjoint subalgebra M L.H / (that is, when 1 2 M and M D M) coincides with its bicommutant M00 if and only if it is closed in the strong operator topology or if and only if it is closed in the weak operator topology. The algebras M satisfying one of these conditions are called von Neumann algebras. The self-adjoint subalgebras N L.H / closed in the uniform topology are called C -algebras. The class of C -algebras is signiﬁcantly wider than that of von Neumann algebras. A commutative von Neumann algebra M (that is, where AB D BA for all A, B 2 M) is called maximal if there are no commutative von Neumann algebras strictly containing M. A von Neumann algebra M is called a factor if its center is isomorphic to C. A projection P 2 L.H / is any self-adjoint idempotent, P D P D P 2. The bicommutant theorem implies that any von Neumann algebra contains a multitude of projections and is, in fact, fully determined by its projections. This differs markedly with C -algebras which may not contain any nontrivial projections at all. A self-adjoint operator A 2 L.H / is called positive if hA, i0forall 2 H . The collection of all positive elements of L.H / is denoted by L.H /C.Thissetisa proper closed cone in L.H / and it induces a partial ordering in the set of all self-adjoint operators from L.H / by setting A B if and only if B A 2 L.H /C.Foragiven von Neumann algebra M we set MC :DfA 2 M : A 0g, which is called the positive part of M. Vigier’s theorem states that von Neumann algebras have the least upper bound property.

Theorem 2.1.1 (Vigier’s Theorem). If fAi gi2I is an increasing net in MC, bounded from above by B in L.H /, then there exists A 2 MC such that Ai " A strongly and A B. Let Proj.M/ denote the projections in a von Neumann algebra M.ThesetProj.M/ is a lattice, with supremum and inﬁmum given by the following procedure. Let p and q be any two projections from M. There exists a projection, denoted by p _ q,that is greater than p and q, but it is less than or equal to any projection r 2 M such that r p and r q. Similarly, there exists a projection, denoted by p ^ q,thatisless than or equal to p and less than or equal to q, yet it is greater than or equal to any projection r 2 M such that r p and r q.Thatis,

p _ q :D inffr 2 Proj.M/ : r p, r qg p ^ q :D supfr 2 Proj.M/ : r p, r qg.

The following, motivating, example of a maximal commutative von Neumann algebra indicates that the lattice structure generalizes set union and set intersection. 40 Chapter 2 Preliminaries on Symmetric Operator Spaces

Example 2.1.2. Let .X, †, / be a -ﬁnite measure space. Let L2.X/ denote the Hilbert space of all (equivalence classes of) square integrable functions on X, and let L1.X/ denote the algebra of all (equivalence classes of) bounded functions on X.

(a) The mapping f ! Mf is an injective -homomorphism from L1.X/ into L.L2.X//. Here, the operator of pointwise multiplication Mf is given by the formula

.Mf x/.t/ D f.t/x.t/, x 2 L2.X/.

(b) L1.X/ is a maximal commutative von Neumann subalgebra of L.L2.X//.

(c) Every set A 2 † generates a projection A 2 L1.X/. Moreover, for every projection p 2 L1.X/, there exists a set A 2 † such that p D A.

(d) If p D A and q D B , then

p _ q D A[B , p ^ q D A\B .

That is, the operations _, ^ on the Boolean algebra of projections Proj.L1.X// correspond to the operations [, \ on the -algebra †.

(e) If X D .0, 1/ or .0, 1/, † is the -algebra of Lebesgue measurable sets and is Lebesgue measure, then L1.X/ D L1.0, 1/ or L1.X/ D L1.0, 1/.

ZC (f) If X D ZC, † D 2 and is a counting measure, then L1.X/ D l1. Example 2.1.2 is especially important because of the following theorem.

Theorem 2.1.3. A commutative von Neumann algebra on a separable Hilbert space H is isometrically isomorphic to L1.X/ for some measure space .X, †, /. That the lattice of projections Proj.M/ is an extension of the concept of a -algebra follows from Vigier’s theorem: Proj.M/ is a complete sublattice of Proj.L.H // [227, p. 41, 69], that is _ pi 2 Proj.M /, i2I and ^ pi 2 Proj.M /, i2I for any net of projections fpi gi2I Proj.M /. A projection p contains the projection q if q p (or p ^ q D q). A von Neumann algebra M is atomic if every non-zero projection in M contains a non-zero minimal projection and atomless if there are no Section 2.1 Von Neumann Algebras 41

non-zero minimal projections in M. The algebra L1.RC/ in Example 2.1.2 (e) is atomless, while the algebra L1.ZC/ in Example 2.1.2 (f) is atomic. A projection p 2 Proj.M/ is said to be Abelian if the reduced von Neumann algebra pMp is commutative.

1 Example 2.1.4. Let fengnD0 be any orthonormal basis of a separable Hilbert space H . The rank one projections pn :D en ˝ en : !h, enien, n 0, described in Chapter 1, are minimal projections in L.H /, making L.H / an atomic von Neumann algebra. Since pnL.H /pn C, then each pn is an Abelian projection.

The -algebra of all Borel subsets of C is denoted by B.C/. A projection valued measure E is a map E : B.C/ ! Proj.M/ such that E.C/ D 1 and such that the map

F !hE.F /, i, F 2 B.C/ is a complex-valued Borel measure for each , 2 H . If A is a compact normal operator with eigenvalue 2 .A/,letEA./ be the projection onto the eigenspace of .DeﬁneEA as the projection valued measure as- signing the projection EA./ to the points 2 .A/ and the zero projection to all other points in C. Then the spectral theorem for compact operators is equivalent to the statement Z

A D dEA./, C which is shorthand notation for Z

hA, iD dhEA./, i, , 2 H . C

A similar result exists for bounded normal operators.

Theorem 2.1.5 (Spectral Theorem). If A 2 M is a normal operator, then there exists a uniquely determined projection valued measure

EA : B.C/ ! Proj.M/ supported on .A/ such that Z

A D dEA./. C 42 Chapter 2 Preliminaries on Symmetric Operator Spaces

If f is a Borel function bounded on .A/ then, Z

f.A/:D f./dEA./ C deﬁnes a bounded operator f.A/ 2 M.

The projection valued measure EA is called the spectral measure of A and the projections EA.F /, F 2 B.C/, are called the spectral projections of A.IfA is self-adjoint (respectively, positive), then the projection valued measure EA is supported on R (respectively, RC). For a self-adjoint operator A 2 M,set

AC D AEAŒ0, 1, A DAEA.1,0/.

Since A D AC A, it follows that every self-adjoint operator is a linear combination of two positive operators. Every operator A 2 M can be uniquely represented as a linear combination of two self-adjoint operators by the formula A D

1 1

2.2 Semiﬁnite Normal Traces

Assume that M is a von Neumann algebra.

Deﬁnition 2.2.1. Amap : MC ! Œ0, 1 is said to be a semiﬁnite trace if (a) is additive, that is

.˛1A1 C ˛2A2/ D ˛1.A1/ C ˛2.A2/

for all positive numbers ˛1, ˛2 and all positive elements A1, A2 2 MC.

(b) is semiﬁnite, that is for each projection p 2 M there exists an increasing net of projections pi 2 M, i 2 I , such that .pi /<1, i 2 I , and such that p D_i2I pi . Section 2.2 Semiﬁnite Normal Traces 43

(c) is a trace, that is .U AU / D .A/ for all positive elements A 2 MC and all unitary elements U 2 M. This definition of a trace is different to Definition 1.2.10 since we have indicated that is a function on MC that may take extended positive values, and not specifically talked about it being a linear functional on a two-sided ideal. We will first talk about semifinite traces, specifically normal semifinite traces, in the standard fashion and later, in Section 2.7, we discuss traces on bimodules of von Neumann algebras (which will be the equivalent notion to Definition 1.2.10). If : MC ! Œ0, 1 is additive then a projection p 2 Proj.M/ such that .p/ < 1 is called -finite. An equivalent condition for to be semifinite is that M is generated by -finite projections. If .1/<1 then is called finite. A trace is said to be faithful if A 2 MC and .A/ D 0 implies that A D 0. The following condition on a trace corresponds to additivity in measure theory.

Deﬁnition 2.2.2. A semiﬁnite trace : MC ! Œ0, 1 is said to be normal if, for any increasing net Ai 2 MC, i 2 I , with Ai " A, we have

.A/ D sup .Ai /. i2I

Recall that M is represented on a separable Hilbert space. In this case, M is - ﬁnite, that is, every set of pairwise orthogonal projections is at most countable [243, Proposition 3.19]. To see the analogy to countable additivity consider a countable 1 family fpkgkD0 of disjoint projections. We have _1 X1 pk D .pk/ . kD0 kD0

The analogy with an integral in measure theory is seen by using the fact that any A 2 MC is strongly approximatedP by linear combinations of disjoint -ﬁnite projections. 1 More precisely, A D kD0 akpk with .pk/<1,and X1 .A/ D ak.pk/. kD0

Example 2.2.3. Let .X, †, / be a -ﬁnite measure space and let L1.X/ be the von Neumann algebra constructed in Example 2.1.2. The integral : L1.X/C ! Œ0, 1 Z

.Mf / :D fd,0 f 2 L1.X/ X is a faithful normal semifinite trace on L1.X/. The integral is finite if is finite.

ZC (a) If X D ZC, † D 2 and is the counting measure, then

X1 : f Dffngn0 ! fn. nD0

(b) If X D .0, 1/, † is the -algebra of Lebesgue measurable sets and is Lebesgue measure, then Z 1 : f ! f.s/ds 0 is a ﬁnite trace.

(c) If X D .0, 1/, † is the -algebra of Lebesgue measurable sets and is Lebesgue measure, then Z 1 : f ! f.s/ds 0 is a semiﬁnite inﬁnite trace.

Note, however, that L1.X/ is always a ﬁnite von Neumann algebra.

Example 2.2.4. The canonical trace Tr constructed in Section 1.3.1 is finite on every finite rank operator. The set of finite rank operators is strongly dense in the algebra Section 2.2 Semifinite Normal Traces 45

L.H /. It follows that Tr is a faithful, normal and semiﬁnite trace on the von Neumann algebra L.H /. Thus, the pair .L.H /,Tr/ forms an atomic von Neumann algebra with faithful normal semiﬁnite trace.

Example 2.2.5. Fix an orthonormal basis in the separable Hilbert space H . The set of diagonal operators in L.H / (Example 1.1.11 in Chapter 1) is a von Neumann algebra. It is naturally identiﬁed with the von Neumann algebra l1 (considered in Ex- ample 2.1.2 (f)). The restriction of the canonical trace Tr on the diagonal operators is equivalent to the trace introduced in Example 2.2.3 (a).

Example 2.2.6. Every positive trace on a two-sided ideal J of compact operators, by which we mean a unitarily invariant positive linear functional on J as in Section 1.2, is a semifinite trace on the von Neumann algebra L.H /. This follows since the finite rank operators are the minimal two-sided ideal of compact operators and the finite rank projections generate L.H /. Thus a Dixmier trace Tr! from Section 1.3.2, which is finite on the ideal M1,1 of compact operators, is a semifinite trace on L.H /; it is not normal however. Consider the increasing sequence of finite rank operators 1 n An :D diag f.1 C k/ gkD0 which strongly converges to 1 1 A :D diag f.1 C k/ gkD0 . Then

Tr!.An/ D 0, n 0, but

Tr!.A/ D 1.

Hence An ! A but Tr!.An/ 6! Tr!.A/, and Tr! is not normal. The canonical trace Tr is known to be the unique faithful normal semiﬁnite trace on L.H / (up to a scalar multiple) [63]. If is ﬁnite we always assume that .1/ D 1.

Remark 2.2.7. If M is atomic, then we always assume that .p/ D 1 for any atom (minimal projection) p 2 M. It follows that, if p 2 M is a projection, then either .p/ 2 ZC or .p/ D1. Conversely, for every n 2 ZC, there exists a projection p 2 M such that .p/ D n. Note a similar property for the projections of an atomless semiﬁnite von Neumann algebra (see [165] and [166]). 46 Chapter 2 Preliminaries on Symmetric Operator Spaces

(a) If M is a ﬁnite atomless von Neumann algebra with normal ﬁnite trace , then, for every t 2 .0, 1/, there exists a projection p 2 M such that .p/ D t.

(b) If M is an atomless von Neumann algebra with an inﬁnite normal semiﬁnite trace , then, for every t 2 .0, 1/, there exists a projection p 2 M such that .p/ D t.

2.3 Generalized Singular Value Function

In this section we explain the singular value function associated to a semifinite faithful normal trace, and an affiliated space of measurable operators. The singular value function is one of the central concepts in singular trace theory: it provides the Calkin correspondence between Calkin function spaces and bimodules of measurable operators (see Section 2.4 below). The singular value function generalizes the singular value sequence of a compact operator. The reader familiar with the theory of singular values of compact operators presented in such books as Gohberg and Krein [98,99], and Simon [222], will find the material in this section a natural extension of the atomic case. By Theorem 2.1.3, elements of commutative von Neumann algebras can be treated as bounded measurable functions. In noncommutative integration theory, elements of semifinite von Neumann algebras are the substitute for bounded measurable functions. The following definition introduces an analog of an unbounded measurable function.

Definition 2.3.1. Let H be a Hilbert space and M L.H / be a von Neumann algebra equipped with a faithful normal semifinite trace . A closed and densely defined operator A :Dom.A/ ! H is said to be (a) affiliated with the von Neumann algebra M if it commutes with every unitary operator from the commutant of M;

(b) -measurable if it is afﬁliated with M and if, for every >0, there exists a projection p 2 M such that .1 p/H is contained in the domain Dom.A/ of A and .p/ . The set of all -measurable operators is denoted by S.M, /.

It should be noted that M S.M, /. The reader unfamiliar with unbounded operators can understand this section by thinking of elements of M in the place of S.M, /.

Example 2.3.2. If M D L.H / and D Tr, then every -measurable operator A is bounded, that is, S.L.H /,Tr/ D L.H /.IfM is a finite von Neumann algebra equipped with a faithful normal finite trace , then every closed operator affiliated with M is -measurable. Section 2.3 Generalized Singular Value Function 47

Proof. Let A be a closed and densely-defined operator which is Tr-measurable. Taking <1, the only projection p 2 L.H / with Tr.1 p/ is 1. Hence, the domain Dom.A/ of the operator A coincides with H .SinceA : H ! H is a closed operator, it follows from the Closed Graph Theorem that A is bounded. Now, let M be a finite von Neumann algebra and let A be a closed linear operator M affiliated with .Wehave^t>0EjAj.t, 1/ D 0. Since is a normal trace, it follows that

lim .EjAj.t, 1// D 0. t!1

For a given >0, one can take p D EjAj.t, 1/ for sufﬁciently large t. The operator A.1 p/ is bounded and, therefore, .1 p/H is contained in Dom.A/.By Deﬁnition 2.3.1, the operator A is -measurable.

Since Tr is (up to a constant) the unique faithful normal semiﬁnite trace on the von Neumann algebra L.H /, the pair .L.H /,Tr/ is always implicit when we discuss L.H /. The next example is frequently used and also deﬁnes notation.

Example 2.3.3. We indicate what the -measurable operators are for the commutative von Neumann algebras in Example 2.2.3.

(a) If M D l1 with the trace given by the sum as in Example 2.2.3,thenS.M, / D l1.

(b) If M D L1.0, 1/ with the ﬁnite trace given by the integral as in Example 2.2.3, then S.M, / (denoted further by S.0, 1/) consists of all measurable functions on .0, 1/.

(c) If M D L1.0, 1/ with the semifinite infinite trace given by the integral as in Example 2.2.3,thenS.M, / (denoted further by S.0, 1/) consists of all measurable functions x on .0, 1/ whose unbounded support sets have finite measure

mft 2 .0, 1/ : jx.t/jsg < 1

for all sufﬁciently large s. Here, m is Lebesgue measure (compare this with an essentially bounded function where the unbounded support sets have measure zero for all sufﬁciently large s).

The notation S may refer to the space S.0, 1/, S.0, 1/ or S D l1, if the context is clear. The reader can ﬁnd references for the following results in the end notes of this section.

Lemma 2.3.4. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . Suppose that A, B are -measurable operators. 48 Chapter 2 Preliminaries on Symmetric Operator Spaces

(a) A C B, AB and A are -measurable operators. Here, addition and multiplication are understood in the “strong sense”, that is AB is the closure of the product AB (a suitable dense domain so that Deﬁnition 2.3.1 is satisﬁed can be found, see the references in the end notes), and A C B is the closure on the domain Dom.A/ \ Dom.B/. (b) jAjD.AA/1=2 is a -measurable operator. Moreover, there exists a partial isometry U 2 M such that A D U jAj.

Theorem 2.3.5. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . The set S.M, /is an algebra with involution with respect to strong addition and multiplication and the conjugation operation. We deﬁne the generalized singular value function.

Definition 2.3.6. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . For every A 2 S.M, /, the generalized singular value function of A, .A/ : t ! .t, A/, t>0, is defined by the formula

.t, A/ :D inffkA.1 p/k1 : p 2 Proj.M/, .p/ tg.

Remark 2.3.7. The infimum exists. If one takes p as in Definition 2.3.1,thenA.1p/ : H ! H is a closed operator. Thus, by the Closed Graph Theorem, A.1 p/ is bounded and the quantity kA.1 p/k1 is finite. Equivalently, the function .A/ can be defined in terms of the spectral distribution function nA of the operator A 2 S.M, /. Recalling that the von Neumann algebra M contains all the spectral projections of jAj2S.M, /,weset

nA.s/ :D .EjAj.s, 1//, s 0, (2.1) where EjAj denotes the spectral measure of the operator jAj. The singular value function can be expressed as follows

.t, A/ D inffs 0: nA.s/ tg, t>0. (2.2) Direction to the proof of this formula can be found in the end notes.

Remark 2.3.8. Formula (2.2) shows that the function .A/ is the right inverse of the function nA. However, it is not the left inverse, that is

.nA.t/, A/ D inffs 0: nA.s/ nA.t/gt. This fact is used throughout the text. Section 2.3 Generalized Singular Value Function 49

On one hand, the singular value function generalizes the classical notion of decreasing rearrangement [10].

Example 2.3.9. Given a function x 2 L1.0, 1/, deﬁne the decreasing rearrangement t ! .t, x/ by the formula

.t, x/ :D inffs 0: m.fjxj >sg/ tg where m is Lebesgue measure. This might seem like an overlap of notation, but if the algebra L1.0, 1/ is embedded in L.L2.0, 1// so that every x 2 L1.0, 1/ is represented by the multiplication operator Mx : L2.0, 1/ ! L2.0, 1/, then we have .x/ D .Mx /. On the other hand, the singular value function generalizes the singular value sequence of a compact operator.

Example 2.3.10. For the pair .L.H /,Tr/, the function .A/ is a step function and .n, A/, n 0, are the singular values of a compact operator A.

The following embedding result can be found in [234] (see also the forthcoming book [68]). It can be viewed as an extension of the Schmidt decomposition, which is where the commutative sequence space l1 can be embedded in the diagonal of L.H / according to an eigenbasis of a positive compact operator.

Theorem 2.3.11. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace and let 0 A 2 S.M, / be such that .1, A/ D 0. There is a commutative von Neumann subalgebra M0 of M (with A 2 S.M0, /) such that

jM0 is semiﬁnite.

(a) If M is atomic, then M0 can be chosen atomic and -ﬁnite (but inﬁnite).

(b) If M is atomless and .1/ D1, then M0 can be chosen -ﬁnite (but inﬁnite).

Moreover, there exists a -isomorphism i : S.0, 1/ ! S.M0, / (respectively, i : S.0, 1/ ! S.M0, / or i : l1 ! M0) such that .i.x// D .x/ for every x 2 S.0, 1/ (respectively, for every x 2 S.0, 1/ or x 2 l1) and such that i..A// D A. The condition .1, A/ D 0 in the statement of Theorem 2.3.11 is necessary. In- deed, consider the situation M D L1.0, 1/, x.t/ :D arctan.t/, t 0. It is straightforward that .t, x/ D =2, t 0 (see also [139]) and that there is no isomorphism i : S.0, 1/ ! S.0, 1/ such that i..x// D x. The following properties of generalized singular value functions follow almost immediately from the deﬁnition. 50 Chapter 2 Preliminaries on Symmetric Operator Spaces

Lemma 2.3.12. Let A, B 2 S.M, /. (a) The function t ! .t, A/, t>0, is decreasing and right-continuous.

(b) .t, A/ !kAk1 when t ! 0 for bounded A and .t, A/ !1when t ! 0 for unbounded A. (c) .t, A/ D .t, jAj/ for all t>0. (d) If ˛ 2 C,then.t, ˛A/ Dj˛j.t, A/ for all t>0. (e) If 0 B A, then .t, B/ .t, A/ for all t>0. (f) If .1/ D 1, then .t, A/ D 0 for all t>1. The singular values of a ﬁnite rank operator A measure how well the operator can be approximated by operators of rank less than or equal to that of A,thatis

.n, A/ D inffkA Bk1 :rank.B/ ng, n 0.

A similar formula, enhancing that of Deﬁnition 2.3.6, holds for an arbitrary operator A 2 S.M, /.

Theorem 2.3.13. For every operator A 2 S.M, /, we have

.t, A/ D inffkA Bk1 : B 2 S.M, /, nB .0/ tg.

Remark 2.3.14. The inﬁmum exists. If A D U jAj is a polar decomposition of A, then we may take B D U jAjEjAj.n, 1/ for sufﬁciently large n. Then A B is a bounded operator and kA Bk1 exists. We need the following technical result which will be frequently used. The usual triangle inequality for the modulus of complex numbers is no longer valid for the modulus of operators, that is, it may be that

jA C Bj 6jAjCjBj, A, B 2 S.M, /.

The next result is a very useful substitute.

Lemma 2.3.15. Let Ak 2 S.M, /, k 0. There exist partial isometries Uk 2 M, k 0, such that ˇ ˇ ˇ ˇ ˇXn ˇ Xn ˇ ˇ ˇ Ak ˇ Uk jAk jUk, n 0. kD0 kD0 The properties in the following corollary are similar to inequalities of Fan for compact operators [91, 98]. They underly the Calkin correspondence and many other results. In view of their importance, we provide short proofs. Section 2.3 Generalized Singular Value Function 51

Corollary 2.3.16. Let A, B 2 S.M, / and t, s>0. Then .a/.tC s, A C B/ .t, A/ C .s, B/. .b/.tC s, AB/ .t, A/.s, B/.

Proof. The proof of the estimates is very similar. Fix >0. By Theorem 2.3.13, there exists A1 such that

nA1 .0/ t, kA A1k1 .t, A/ C .

Similarly, there exists B1 such that

nB1 .0/ s, kB B1k1 .s, B/ C . (a) We have

k.A C B/ .A1 C B1/k1 .t, A/ C .s, B/ C 2. By Lemma 2.3.15, there exist partial isometries U and V such that jA1 C B1jU jA1jU C V jB1jV .

We infer that nA1CB1 .0/ t C s.Since is arbitrary small, the result follows by using Theorem 2.3.13 again.

(b) Deﬁne an operator X by the formula X :D AB .A A1/.B B1/.Itisclear that

kAB Xk1 Dk.A A1/.B B1/k1 ..t, A/ C /..s, B/ C /.

Note that X D A1.B B1/ C AB1. It follows from Deﬁnition 2.3.6 and the

assumption that nB1 .0/ s that .s, B1/ D 0. Similarly .t, A1/ D 0. By

Deﬁnition 2.3.6 again, we obtain that .s, AB1/ D 0andnAB1 .0/ s. Similarly

arguing we obtain that nA1.BB1/.0/ t.From

nAB1 .0/ s, nA1.BB1/.0/ t,

we infer that nX .0/ t C s.Since is arbitrarily small, the result follows by using Theorem 2.3.13 again.

Some additional properties of the singular value function are summarized in the following corollary.

Corollary 2.3.17. Let A, B 2 S.M, / and t, s>0. (a) If B is bounded, then .t, BA/ kBk1.t, A/, .t, AB/ kBk1.t, A/, .t, A / D .t, A/. 52 Chapter 2 Preliminaries on Symmetric Operator Spaces

(b) The operation A ! .t, A/ is unitarily invariant. That is, .t, A/ D .t, U AU / for every unitary U 2 M.

j.s, A/ .s, B/jkA Bk1, 8A, B 2 M, 8s>0.

(d) If A 0 and if f : RC ! RC is a continuous and increasing function, then .f .A// D f..A//.

Proof. (a) The inequalities follow from Corollary 2.3.16 and the inequality .B/ kBk1.IfA D U jAj is a polar decomposition of A,thenjA jDU jAjU . Therefore, .t, A/ D .t, jAj/ D .t, U jAjU / .t, jAj/ D .t, A/. Substituting A instead of A, we obtain the reverse inequality .t, A/ .t, A/. (b) From (a) we have .t, U AU / .t, A/ since kU k1 DkU k1 D 1. Simi- larly, .t, A/ D .t, U.UAU /U / .t, U AU /. (c) We apply Corollary 2.3.16 and the property .t, A/ !kAk1 as t # 0 for every A 2 M. (d) It is sufﬁcient to establish the equality only for those values of t such that f is not a constant in the neighborhood of t. For such values of t we have, by the spectral theorem, Ef.A/.f .t/, 1/ D EA.t, 1/.

Results that are shown for atomless semiﬁnite von Neumann algebras, and which rely on singular values, can be transferred to arbitrary semiﬁnite von Neumann algebras by the following standard device.

Lemma 2.3.18. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace .

(a) The von Neumann algebra M ˝ L1.0, 1/ equipped with the faithful normal semiﬁnite trace ˝m (where m D is from Example 2.2.3 and m is Lebesgue measure) is atomless.

(b) For every A 2 S.M, /, .A ˝ 1/ (the singular value function of the operator A ˝ 1 with respect to the trace ˝ m) coincides with .A/ (the singular value function of the operator A with respect to the trace ).

Proof. Only the second claim has to be proved. It is clear that EjAj˝1.s, 1/ D EjAj.s, 1/ ˝ 1 for every s>0. Therefore, nA˝1.s/ (with respect to the trace ˝ m) equals nA.s/ (with respect to the trace ). The claim now follows from formula (2.2). Section 2.4 Calkin Correspondence in the Semiﬁnite Setting 53

2.4 Calkin Correspondence in the Semiﬁnite Setting

In this section we extend the Calkin correspondence between two-sided ideals of compact operators and Calkin sequence spaces (explained in Section 1.2) into the semiﬁ- nite setting.

Deﬁnition 2.4.1. A linear subspace J .M, / of S.M, / is called a Calkin operator space if B 2 J .M, / whenever B 2 S.M, / and .B/ .A/ for some A 2 J .M, /. A Calkin function (respectively, sequence) space is the term reserved for a Calkin operator space when M D L1.0, 1/ or M D L1.0, 1/ (respectively, M D l1). Following our convention of denoting function or sequence spaces by straight letters, a Calkin function or sequence space is denoted by J . The symbol Lp, p 1, always refers to the Lebesgue Lp-spaces (either on .0, 1/ or on .0, 1/).

Example 2.4.2. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . (a) It is a triviality that S.M, / is a Calkin operator space.

(b) For every p 1 (including p D1), the linear space Lp is a Calkin function space.

Lp.M, / :DfA 2 S.M, / : .A/ 2 Lpg

is a Calkin operator space. The space Lp.M, / contains unbounded operators from S.M, / if M is atomless.

(d) The closure C0.M, / of the linear span of all -ﬁnite projections in the uniform norm is a Calkin operator space (see Lemma 2.6.9 below).

These examples are discussed in more detail in Section 2.6. In Deﬁnition 2.6.8 the space C0.M, / is called the algebra of -compact operators. The following direct sum operation will be very useful later in the text. We use it in the proof of Theorem 2.4.4 below.

Deﬁnition 2.4.3. Let Ak 2 S.M, /, k 0. If pk 2 M, k 0, are pairwise orthogonal projections, and if Bk 2 pkS.M, /pk are such that .Bk / D .Ak/, k 0, then we write M X Ak :D Bk. k0 k0 54 Chapter 2 Preliminaries on Symmetric Operator Spaces

Observe thatL if the operators Ak , k 0, are taken from a Calkin operator space J M J M . , /,then k0 Ak 2 . , /. The operation introduced in Definition 2.4.3 is always possible whenever M D L1.0, 1/, M D l1,orifM is a semifinite non- finite factor.L The key property of the direct sum operation is that if f :D .A/,then 2f D .A A/.Here2 is the dilation operator. This fact is used repeatedly in the rest of this book and in particular in the next theorem. Theorem 2.3.11 reveals that the singular value function (through the -isomorphism i)isamapbetween-measurable operators of semifinite von Neumann algebras and function spaces of measurable functions (respectively, a map between an atomic von Neumann algebra and the sequence space l1). The following theorem extends the Calkin correspondence between two-sided ideals of L.H / and their Calkin sequence spaces by showing that the singular value function maps bijectively Calkin operator spaces to Calkin function spaces.

Theorem 2.4.4. Let M be an atomless (or atomic) von Neumann algebra equipped with a faithful normal semiﬁnite trace .IfJ .M, / is a Calkin operator space, then

J :Dfx 2 S : .x/ D .A/, A 2 J .M, /g is a Calkin function (or sequence) space. If J is Calkin function (or sequence) space, then J .M, / :DfA 2 S.M, / : .A/ 2 J g is a Calkin operator space. This provides a canonical bijection between Calkin operator spaces and Calkin function (or sequence) spaces.

Proof. Suppose that M is atomless and .1/ D1. We prove this case, with the other cases proved similarly. Let J :D J .M, / be a Calkin operator space and let x 2 J . By deﬁnition, there exists an operator A 2 J with .A/ D .x/.SinceJ is a Calkin operator space, we may assume without loss of generality that A .1, A/ (so that Theorem 2.3.11 is applicable to the operator A .1, A/). Let M0 be the commutative subalgebra constructed in Theorem 2.3.11 and let i be the corresponding isomorphism. Let y 2 S be such that .y/ .x/.IfB D i..y//,then.B/ D .y/ .A/.SinceJ is a Calkin operator space, it follows that B 2 J . Hence, y 2 J . We now prove that J is a linear space. If x, y 2 J ,then.x/ 2 J and .y/ 2 J . Let A D i..x//, B D i..y// and C D A C B.SinceJ is a Calkin operator space, it follows that A, B 2 J . Therefore, C 2 J and C ˚ C 2 J . It follows from the Corollary 2.3.16(a) that

.x C y/ 2..x/ C .y// D .C ˚ C/.

It follows from the previous paragraph that x C y 2 J . Section 2.4 Calkin Correspondence in the Semiﬁnite Setting 55

The proof of the converse assertion is similar and is, therefore, omitted. The functor J ! J is the inverse to the functor J ! J , so the correspondence is bijective.

Theorem 2.4.4 is an extension of the Calkin correspondence given in Theorem 1.2.3 in Chapter 1. This follows because all two-sided ideals of bounded operators are Calkin operator spaces for M D L.H /. We show this result for general factors.

Deﬁnition 2.4.5. A linear subspace J of S.M, / is called an operator bimodule on M if AB, BA 2 J whenever A 2 J and B 2 M.

If J is an operator bimoduleon M and is a subset of M, then it is an ideal of M.The two-sided ideals of bounded operators are all the operator bimodules for M D L.H /. The noncommutative Lp-spaces for atomless non-ﬁnite M that were introduced in Example 2.4.2 provide examples of operator bimodules that contain unbounded operators.

Lemma 2.4.6. If M is a semiﬁnite factor, then every operator bimodule is a Calkin operator space.

Proof. Suppose that M is atomless and that .1/ D1. We prove this case, with the other cases proved similarly. Let 0 B 2 S.M, /and 0 A 2 J be such that .B/ .A/.If.1, B/ D 0, then .1, A/ D 0. We can apply Theorem 2.3.11 to the operators A and B.There M exist commutative subalgebras k, k D 1, 2, such that jMk is semiﬁnite and such that B 2 S.M1, /, A 2 S.M2, /.Letik, k D 1, 2, be the trace-preserving - isomorphisms constructed in Theorem 2.3.11. Choose a step function u such that .B/ u 2.A/ and select functions u1, u2 2 L1.0, 1/ such that .B/ D uu1 and u D 2.A/u2.LetAk D ik.u/, k D 1, 2. Since A2 D 2A i2.u2/ and since J is a bimodule, it follows that A2 2 J . By construction, the operators A1 and A2 have identical discrete spectrum and their corresponding eigenprojections have the same trace. Since M is a factor, it follows that there exists a partial isometry U which conjugates the corresponding eigenprojections of A1 and A2. Consequently, A1 D UA2U and, therefore, A1 2 J .Since B D A1 i1.u1/ and since J is a bimodule, it follows that B 2 J . Let 0 B 2 S.M, / and 0 A 2 J be such that .B/ .A/ and .1, B/ > 1 1 0. Since EA. 2 .1, A/, 1/ commutes with A, it follows that EA. 2 .1, A/, 1/ 2 J . Thus, J contains an inﬁnite projection. Since M is a factor, it follows that 1 2 J . By Corollary 2.3.17(d) ..B .1, A//C/ D ..B/ .1, A//C ..A/ .1, A//C D ..A.1, A//C /. It follows from above that .B .1, A//C 2 J . Thus, B 2 J .

Remark 2.4.7. If the semiﬁnite von Neumann algebra M is not a factor, then the assertion of Lemma 2.4.6 fails. Indeed, take a projection p from the center of M and 56 Chapter 2 Preliminaries on Symmetric Operator Spaces set J D pM. Then J is an operator bimodule on M which fails to be a Calkin operator space. If a semiﬁnite von Neumann algebra M is not atomic or atomless, then associating a Calkin operator space to a Calkin function space J is still valid

J .M, / :DfA 2 S.M, / : .A/ 2 J g, but, in general, the inability to associate the spaces S.M0, / in Theorem 2.3.11 to a ﬁxed function space such as S.0, 1/ (or S.0, 1/, or the sequence space l1) prevents the converse. The most important examples of operator bimodules and Calkin operator spaces are delivered by symmetric operator spaces.

2.5 Symmetric Operator Spaces

Motivated by the Calkin correspondence we now define the main objects of our study: symmetric operator spaces and, in Section 2.7, continuous singular traces on symmetric operator spaces. Let .M, / be the pair of a von Neumann algebra M with a fixed faithful normal semifinite trace .

Deﬁnition 2.5.1. A Banach subspace .E.M, /, kkE / of S.M, / is called a symmetric operator space if, for A 2 E.M, / and for every B 2 S.M, / with .B/ .A/, we have B 2 E.M, / and kBkE kAkE . A symmetric function (respectively, sequence) space is the term reserved for a symmetric operator space when M D L1.0, 1/ or M D L1.0, 1/ (respectively, M D l1). Following our convention of denoting function or sequence spaces by straight letters, a symmetric function or sequence space is denoted by E. If required E.0, 1/ denotes a function space when M D L1.0, 1/ and E.0, 1/ denotes a function space when M D L1.0, 1/. Symmetric function spaces are well-studied in Banach space theory [10, 139, 150]. Evidently, symmetric operator spaces are a noncommutative generalization of symmetric function and sequence spaces. Every symmetric operator space is a Calkin operator space. If E.M, / M is a symmetric operator space, then E.M, / is called a symmetric operator ideal. Due to the Calkin correspondence, the symmetric operator spaces associated to M D L.H / are, except for L.H / itself, two-sided ideals of compact operators. The reader should note that, unless .M, / is appended as in the notation E.M, /,then the notation E with no appended term will denote a symmetric ideal of compact operators E.L.H /,Tr/. Theorem 2.5.2 shows that the proper symmetric operator spaces associated to the factor L.H /, in the terminology introduced above, are exactly the Section 2.5 Symmetric Operator Spaces 57 symmetrically normed ideals of compact operators studied by Gohberg and Krein, and Simon [98, 99, 222].

Theorem 2.5.2. Let M be a semiﬁnite factor. The following statements are equivalent.

(a) .E.M, /, kkE / is a Banach space and

kBAC kE kBk1kAkE kC k1, A 2 E.M, /, B, C 2 M. (2.3)

(b) .E.M, /, kkE / is a symmetric operator space.

Proof. If E.M, / satisfies condition (a), then we can argue as in the proof of Lem- ma 2.4.6. That is, fix 0 B 2 S.M, /and 0 A 2 E.M, /such that .B/ .A/ and select, for a given >0, a partial isometry U such that jBj.1 C /UjAjU . Since >0 is arbitrarily small, it follows that B 2 E.M, / and kBkE kAkE . If E.M, / satisfies the condition (b), then the condition (a) follows from the inequality .BAC / kBk1kC k1.A/.

The next result shows that the Calkin correspondence, as in the last section, restricts to a bijective correspondence between symmetric operator spaces and symmetric function spaces. It also shows that the correspondence may be seen as isometric embedding of a function (or sequence) space into its operator counterpart.

Theorem 2.5.3. Let M be a semiﬁnite von Neumann algebra and let E.M, / be a symmetric operator space. (a) If M is atomless and if .1/ D1, then set

E :Dfx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g.

(b) If M is atomless and if .1/ D 1, then set

E :Dfx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g.

E :Dfx 2 l1 : .x/ D .A/ for some A 2 E.M, /g.

Set

kxkE :DkAkE for all x 2 E, A 2 E.M, / such that .x/ D .A/.

So deﬁned .E, kkE / becomes a symmetric function (or sequence) space. 58 Chapter 2 Preliminaries on Symmetric Operator Spaces

Proof. Let M0 be the commutative von Neumann algebra constructed in Theo- rem 2.3.11. Set E :D E.M, / \ S.M0, /. Observe that E is a symmetric operator space of M0 when equipped with kkE . It follows that one can consider E as a symmetric function space on .0, 1/ (respectively, function space on .0, 1/ or sequence space).

The primary question in symmetric operator space theory is whether the converse to Theorem 2.5.3 holds (given a function or sequence space associate to it an operator space), so that the Calkin correspondence between a symmetric operator space and a symmetric function space is, in fact, an isometric correspondence between Banach spaces. The question goes back to J. von Neumann in 1937 [172].

Question 2.5.4. Let E be a symmetric function (respectively, sequence) space and let M be a semiﬁnite atomless (respectively, atomic) von Neumann algebra. Set

E.M, / :DfA 2 S.M, / : .A/ 2 Eg, kAkE :Dk.A/kE .

Is it true that so deﬁned .E.M, /, kkE / is a symmetric operator space? Given that a Calkin operator space can be associated to any symmetric function space, as mentioned in Section 2.4, one can ask a stronger question.

Question 2.5.5. Let E be a symmetric function space on .0, 1/ and let M be a semiﬁ- nite von Neumann algebra. Set

E.M, / :DfA 2 S.M, / : .A/ 2 Eg, kAkE :Dk.A/kE .

Is it true that so defined .E.M, /, kkE / is a symmetric operator space? We answer these questions in the affirmative in Part II, where we also discuss the importance of the affirmative answer to transferring between functionals on symmetric function spaces and traces on symmetric operator spaces. The operator spaces of interest to us in the applications will satisfy a stronger symmetric property and possess a Fatou norm. A symmetric operator space is fully symmetric if we replace the order in Definition 2.5.1 by submajorization . Subma- jorization is a fundamental concept in classical symmetric function space theory, and we discuss it in Section 3.3. When M is the algebra l1 the following definition reduces to the classical notion of submajorization of sequences.

Deﬁnition 2.5.6. Let M be a semiﬁnite von Neumann algebra and let A, B 2 S.M, /. The operator B is said to be submajorized by A and written B A if Z Z t t .s, B/ds .s, A/ds, t 0. 0 0 Section 2.6 Examples of Symmetric Operator Spaces 59

Deﬁnition 2.5.7. A symmetric operator space E.M, / is called

(a) strongly symmetric, if for A, B 2 E.M, /with .B/ .A/, we have kBkE kAkE ,

(b) fully symmetric, if for A 2 E.M, / and every B 2 S.M, / with .B/ .A/, we have B 2 E.M, / and kBkE kAkE .

If, in the deﬁnition above, the symmetric operator space is a function or sequence space, then we speak of a fully symmetric function or sequence space. A symmetric operator space E.M, / is strongly symmetric if and only if it is a closed subspace of a fully symmetric operator space. To see this, deﬁne F as a subset of S.M, / by letting B 2 F if and only if there exists A 2 E.M, / such that .B/ .A/ and set

kBkF :D inffkAkE : A 2 E.M, /, .B/ .A/g.

It is easy to verify that F is then a fully symmetric operator space containing E.M, / as a closed subspace. The following property is widely used in the theory of symmetric function spaces and symmetric operator spaces. Furthermore, in some standard references on the subject (e.g. Simon’s book [222]), the requirement that the symmetric space has a Fatou norm appears to be a part of the deﬁnition. We use this property in Chapter 4.

Deﬁnition 2.5.8. The norm kkE on a symmetric operator space E.M, /is a Fatou norm if the unit ball of E.M, / is closed with respect to convergence in the strong operator topology.

In the atomless (or atomic) case all symmetric operator spaces with a Fatou norm are strongly symmetric.

Theorem 2.5.9. If M is an atomless (or atomic) semiﬁnite von Neumann algebra, then every symmetric operator space of M that is equipped with a Fatou norm is a closed subspace of a fully symmetric operator space (that is, strongly symmetric).

Proof. The assertion follows from Theorem 2.5.3 and classical results on symmetric function spaces (see e.g. [10, 139]).

2.6 Examples of Symmetric Operator Spaces

We provide some examples of symmetric operators spaces. The noncommutative Lp operator spaces are well known and extensively studied, as are the so-called -compact operators. Finally, the Lorentz operator spaces and ideals are introduced. 60 Chapter 2 Preliminaries on Symmetric Operator Spaces

Noncommutative Lp-spaces

Recall that the symbol Lp refers to the Lebesgue Lp-spaces (either on .0, 1/ or on .0, 1/). We now add to Example 2.4.2.

Example 2.6.1 (noncommutative Lp-space). Let M be a semiﬁnite von Neumann algebra. For every p 1 (including p D1), deﬁne a linear space

Lp.M, / :DfA 2 S.M, / : .A/ 2 Lpg.

When equipped with the norm kAkp :Dk.A/kp, the operator space Lp.M, / becomes a fully symmetric operator space.

Proof. The noncommutative Lp-space is a Banach space [61, 140, 168, 217]. The Lebesgue Lp-spaces are fully symmetric function spaces. Suppose .B/ .A/. If .A/ 2 Lp,then.B/ 2 Lp and kBkp kAkp.

According to the deﬁnition of the noncommutative L1-space, we have

A 2 L1.M, / , .A/ 2 L1.

This observation, coupled with another straightforward equivalence

A 2 M , .A/ 2 L1 identiﬁes the noncommutative L1-space with M,

M D L1.M, / and justiﬁes the notation for the operator norm on M,

kAk1 Dk.A/k1 D supt2.0,1/.t, A/ D .0, A/.

Generally Lp.M, /,1 p<1, is not an ideal of M. This observation can be further specialized as follows.

Example 2.6.2. Let M be a semiﬁnite von Neumann algebra and 1 p<1.

(a) If M is atomless and .1/ D1,thenLp.M, /corresponds uniquely, amongst Calkin operator spaces, to the fully symmetric function space Lp.0, 1/.Since Lp.0, 1/ 6 L1.0, 1/,thenLp.M, / is not an ideal of M.

(b) If M is atomless and .1/ D 1,thenLp.M, / corresponds uniquely, amongst Calkin operator spaces, to the fully symmetric function space Lp.0, 1/. Hence,

M Lp.M, / L1.M, /. Section 2.6 Examples of Symmetric Operator Spaces 61

(c) If M is atomic (see Remark 2.2.7), then Lp.M, /corresponds uniquely, amongst Calkin operator spaces, to the fully symmetric sequence space lp. Hence,

L1.M, / Lp.M, / M

and the noncommutative Lp-spaces are ideals of M. The following theorem shows that the faithful normal semiﬁnite trace can be extended to the symmetric operator space L1.M, /.

Theorem 2.6.3. The semiﬁnite trace uniquely extends to a continuous linear unitarily invariant functional on L1.M, /. For every positive A 2 L1.M, /, we have Z 1 .A/ D .s, A/ds. (2.4) 0 This functional is normal, that is, for every increasing bounded net of positive operators Ai 2 L1.M, /, i 2 I , we have

.A/ D sup .Ai /. i2I

Proof. Due to Lemma 2.3.18, we may assume without loss of generality that M is a semiﬁnite atomless von Neumann algebra. Let A 2 M be a positive operator such that .1, A/ D 0. Construct the commutative subalgebra M0 and -isomorphism i as in Theorem 2.3.11. For every t>0, pt D i.Œ0,t/ is the projection such that .pt / D t. We have (see e.g. [70, 174]) Z 1 A D i..A// D .t, A/dpt . 0 For a given n 2 N,set

4n1 X k C 1 A D , A .p n p n /. n 2n .kC1/=2 k=2 kD0

Then An, n 2 N, is an increasing sequence and An " A strongly. Since is a normal trace, it follows that .An/ ! .A/ as n !1. By the Levi theorem, we have that

n Z 4X1 1 n k C 1 .An/ D 2 n , A ! .t, A/dt, n !1. 2 0 kD0

This proves the equality (2.4) for such an operator A. The right-hand side of the equation (2.4) is kAk1. 62 Chapter 2 Preliminaries on Symmetric Operator Spaces

Let A D A 2 M be such that .A/ 2 L1.WehaveAC, A 2 M and .AC/, .A/ 2 L1. It follows from the above that

.A/ D .AC/ .A/ .AC/ kACk1 kAk1.

By the Hahn–Banach theorem, there exists an extension of to the Hermitian part of L1.M, / preserving the inequality kk1. However, the ideal fA 2 M : .A/ 2 L1g is dense in L1.M, /. Thus, such an extension of to the Hermitian part of L1.M, / is unique. By linearity, uniquely extends to L1.M, / as a bounded functional. Since is unitarily invariant, it follows that an extension : L1.M, / ! C is unitarily invariant. For every positive A 2 M with .A/ 2 L1,wehave.A/ D kAk1. By continuity, we have .A/ DkAk1 for every positive A 2 L1.M, /.

The duality L1.M, / ! L1.M, / given by A 2 L1.M, / ! .A/ 2 L1.M, /

is an isometry [61, 140, 168, 217]. This identifies the Banach dual L1.M, / of the operator bimodule L1.M, / as the von Neumann algebra M, which is another way to characterize a von Neumann algebra (as the Banach dual of a Banach space). Then L1.M, / is the pre-dual of the von Neumann algebra M. These correspondences are a direct analogy of the Radon–Nikodym theorem for finite measures. The noncommutativeL1-space gives the appropriate notion of “integrable” or “trace class” for each of the examples of semifinite von Neumann algebras in Section 2.1.

Example 2.6.4. For the von Neumann algebra L.H / the noncommutative L1-space L1.L.H /,Tr/ is the ideal of trace class operators L1 (the closure of ﬁnite linear combinations of rank one-projections in the norm Tr.jj/, as in Section 1.3.1. The trace class operators are associated to the pre-dual of L.H / through the duality

L L .H / ! 1 given by B ! Tr.B/, B 2 L.H /.

This is a noncommutative extension of the duality

l1 ! l1 Section 2.6 Examples of Symmetric Operator Spaces 63 given by X1 1 1 b DfbngnD0 ! b.a/ :D anbn, a DfangnD0 2 l1, nD0 which can be seen by restricting to diagonal operators in L.H /.

Example 2.6.5. If .X, / is a measure space and L1.X/ and the integral are as in Example 2.1.2, then the noncommutative L1-space L1.L1.X/, / is the Lebesgue function space L1.X/ of -integrable functions. The duality

L1.X/ ! L1.X/ given by Z

f 7! .Mf / D f d, f 2 L1.X/ X is an equivalent statement of the Radon–Nikodym theorem for ﬁnite measures absolutely continuous with respect to .

We deﬁne the intersection and the Banach sum of the noncommutativeL1-space and L1-space. Let L1 CL1 be the sum of the Banach spaces .L1, kk1/ and .L1, kk1/. Observe that such a sum can be easily deﬁned since both Banach spaces in question are subsets of S.

Definition 2.6.6. Let M be a semifinite von Neumann algebra. Define linear spaces

.L1 C L1/.M, / :DfA 2 S.M, / : .A/ 2 L1 C L1g,

.L1 \ L1/.M, / :DfA 2 S.M, / : .A/ 2 L1 \ L1g.

When equipped with norms Z 1

kAkL1CL1 :D .s, A/ds, kAkL1\L1 :D maxfkAk1, kAk1g, 0 they become Banach spaces.

The Banach sum is a symmetric operator space of M and the intersection .L1 \ L1/.M, / is a symmetric operator ideal of M. From classical results on symmetric function spaces, every symmetric operator space on an atomic or atomless semiﬁ- nite von Neumann algebra is continuously embedded in the symmetric operator space .L1 C L1/.M, / and continuously contains the symmetric operator space .L1 \ L1/.M, /. This observation can be further specialized as follows.

Example 2.6.7. Let M be a semiﬁnite von Neumann algebra and let E.M, / be a symmetric operator space. 64 Chapter 2 Preliminaries on Symmetric Operator Spaces

(a) If M is atomless and .1/ D1, then

.L1 \ L1/.M, / E.M, / .L1 C L1/.M, /.

(b) If M is atomless and .1/ D 1, then

M E.M, / L1.M, /.

L1.M, / E.M, / M.

All the embeddings are continuous.

Proof. We only prove (a). Let E be the symmetric function space constructed in The- orem 2.5.3. It follows from Theorem II.4.1 of [139] that

L1 \ L1 E L1 C L1 with continuous embeddings. The assertion now follows from the equality

E.M, / DfA 2 S.M, / : .A/ 2 Eg and Theorem 2.4.4.

A Class of Compact Operators in the Semifinite Setting If H is a separable Hilbert space, then a projection p 2 L.H / is finite rank if and only if Tr.p/ < 1. In particular, the ideal of compact operators C0.H / is the norm closure of the linear span of projections in L.H / with finite trace.

Definition 2.6.8. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace .Thesetof-compact operators, C0.M, /, is the closure in the norm kk1 of the linear span of -finite projections. This notion is a direct generalization of the ideal of compact operators on a Hilbert space H .If is finite, then every projection is -finite and, therefore, C0.M, / D M. The next lemma shows that C0.M, /is a proper fully symmetric operator ideal when is infinite.

Lemma 2.6.9 (-compact operators). The space of -compact operators is associated to the ideal of essentially bounded functions vanishing at inﬁnity. That is,

C0.M, / DfA 2 S.M, / : .A/ 2 L1, .1, A/ D 0g . Section 2.6 Examples of Symmetric Operator Spaces 65

Proof. Let A 2 M and let .1, A/ D 0. Let A D U jAj be the polar decomposition for A.Foragivenn 2 N,deﬁneAn 2 M by setting Xn1 kkAk1 kkAk1 .k C 1/kAk1 A :D U Ej j , . n n A n n kD0

By the spectral theorem, An ! A uniformly in M.SinceAn 2 C0.M, /, it follows that A 2 C0.M, /. Conversely, A 2 C0.M, / and let An, n 2 N, be a ﬁnite linear combination of - M ﬁnite projections such that An ! A in .Ifan D .EAn .0, 1//,then.t, An/ D 0 for t an. It follows from the Corollary 2.3.17 that

.1, A/ .an, A/ Dj.an, A/ .an, An/jkA Ank1.

Since An ! A in M, it follows that .1, A/ D 0.

Lorentz Operator Spaces Lorentz operator spaces provide further examples of fully symmetric operator spaces. These symmetric operators spaces prove to be fundamental in the theory of singular traces.

Example 2.6.10 (Lorentz operator spaces). Let : RC ! RC be an increasing concave function. The Lorentz function spaces M and M (note that the traditional notation for M is ƒ ) are deﬁned by setting Z 1 t M .0, 1/ :Dfx 2 S.0, 1/ :sup .s, x/ds < 1g t>0 .t/ 0 and Z 1 M .0, 1/ :Dfx 2 S.0, 1/ : .t, x/d .t/ < 1g. 0 Deﬁne the corresponding Lorentz operator spaces by setting

M .M, / :DfA 2 S.M, / : .A/ 2 M .0, 1/g and M M S M . , / :DfA 2 . , / : .A/ 2 M .0, 1/g.

Equipped with the norms Z 1 t kAkM :D sup .s, A/ds, t>0 .t/ 0 66 Chapter 2 Preliminaries on Symmetric Operator Spaces and Z 1 kAkM :D .t, A/d .t/, 0 the Lorentz operator spaces become fully symmetric operator spaces. For the pair .L.H /,Tr/, the symmetric operator spaces M :D M .L.H /,Tr/ M M L and :D . .H /,Tr/ are symmetric ideals of compact operators, and, in fact, are the Lorentz ideals of compacat operators associated to in Example 1.2.7. Observe that the unit ball in M .M, / consists of those A 2 S.M, / with 0 0 .A/ . So, when 2 L1.RC/,thenM .M, / M. A Lorentz operator space satisfying this condition is called a Lorentz operator ideal.

The reader should note that, unless .M, /is appended as in the notation M .M, /, then the notation M with no appended term will always denote the Lorentz ideal of compact operators M .L.H /,Tr/. For the studies of symmetric function (respectively, operator) spaces which generalize classes of Lp-spaces and Lorentz M -spaces we refer to [10, 139] (respectively, [68]). Not every symmetric operator space is fully symmetric. To show such a distinction it is sufﬁcient to show that there is a symmetric function space which is not fully symmetric.

Example 2.6.11. The least symmetric subspace (closed in the norm kkM ) contain- 0 ing the element of a Lorentz space M D M .0, 1/ with the function satisfying the condition .2t/ lim D 1 (2.5) t!0 .t/ does not coincide with M and is, therefore, not fully symmetric. The details can be found in [139, Lemma II.5]. The preceding example is an example of a strongly symmetric function space. Sym- metric function spaces that are not even strongly symmetric are mentioned in the end notes.

Normal Functionals and Trace Duality for Lorentz Operator Spaces

Let .E.M, /, kkE / be a symmetric operator space and let ' belong to the Banach dual E.M, /. Recall that ' is bounded in the dual norm

k'kE :D sup j'.A/j. kAkE1

All the -ﬁnite projections in M belong to E.M, / since .L1 \ L1/.M, / E.M, /. Denote by E.M, /0 the closure of the linear span of -ﬁnite projections Section 2.6 Examples of Symmetric Operator Spaces 67

in M in the norm kkE .ThenE.M, /0 is a symmetric operator space called the separable part of E.M, /(also termed the regular part in Simon [222]). The condition of continuity makes the following equivalent statements evident.

Lemma 2.6.12. For a continuous linear functional ' 2 E.M, / the following statements are equivalent. (a) ' vanishes on the set of -ﬁnite projections in M.

(b) ' vanishes on .L1 \ L1/.M, /.

Deﬁnition 2.6.13. A linear functional ' 2 E.M, / is called

(a) normal if fX˛g˛2I E.M, / and X˛ # 0 implies that '.X˛/ ! 0.

(b) completely additive if A 2 E.M, / and fp˛g˛2I Proj.M/ with p˛ # 0 implies that '.Ap˛/ ! 0 and '.p˛A/ ! 0.

(c) singular if ' vanishes on E.M, /0. E M E M Where required we use the notation . , /n and . , /s for the sets of all normal and singular functionals from E.M, / . Evidently, if E.M, / D E.M, /0, then E.M, / admits no nontrivial singular functional. The following theorem (established in [71]) extends the notion of trace duality, as seen earlier between L1.M, /and the set of normal linear functionals on L1.M, /, to Lorentz operator spaces.

Theorem 2.6.14. For ' 2 M .M, / the following statements are equivalent. (a) ' is normal.

(b) ' is completely additive. M M (c) There exists A 2 . , / such that

'.X/ D .AX/, 8X 2 M .M, /,

and k'kM DkAkM .

The reader may think about the Lorentz operator space M .M, / and its trace M M L M duality with . , / as a generalization of 1. , / and its trace duality with L1.M, /. This result is a generalization of the Radon–Nikodym theorem, and iden- tiﬁes all continuous normal linear functionals on M .M, /. 68 Chapter 2 Preliminaries on Symmetric Operator Spaces

The study of non-normal continuous linear functionals on Lorentz spaces reduces to the study of singular continuous linear functionals. We cite the following special case of the famous Yosida–Hewitt decomposition for more general fully symmetric spaces (see e.g. [64, 72, 73]).

Theorem 2.6.15. Every ' 2 M .M, / admits a unique decomposition

' D 'n C 's , 'n, 's 2 M .M, / (2.6) M M where 'n is normal, in particular, there exists A 2 . , / such that 'n.X/ D .AX/, for all X 2 M .M, /, and 's is singular. The new aspect with Lorentz operator spaces is that continuous singular linear functionals exist on Lorentz operator spaces. Instead of the integration theory based on the normal trace on L1.M, /, we can replace with a continuous singular trace ' 2 M .M, / (as deﬁned in the next section) instead. Prompted by the noncommutative geometry of Alain Connes the couple .M .M, /, '/ replaces the couple .M, / as the basis of a new noncommutative integration theory.

2.7 Traces on Symmetric Operator Spaces

We define a trace on a symmetric operator space in analogy with a trace on a two-sided ideal of compact operators. Let M be a von Neumann algebra equipped with a fixed faithful normal semifinite trace . Recall that every symmetric operator space is an operator bimodule on M.

Definition 2.7.1. Let .E.M, /, kkE / be a symmetric operator space. A unitarily invariant functional ' 2 E.M, / is called a continuous trace. That is, '.U AU / D '.A/ for all A 2 E.M, / and unitaries U 2 M. Since a symmetric operator space E.M, /contains the set of -finite projections of the von Neumann algebra M, a positive trace (we have not requested that a continuous trace be positive in the above definition) on a symmetric operator space E.M, / is, according to the definitions in Section 2.2, a semifinite trace on M. We have already seen one important example of a continuous trace on a symmetric operator space.

Example 2.7.2. The noncommutative L1-space L1.M, / is a symmetric operator space and the extension of the faithful normal semiﬁnite trace

: L1.M, / ! C is a continuous trace on the noncommutative L1-space. If M is a factor, then is the unique normal trace (up to a constant) on L1.M, /. Section 2.7 Traces on Symmetric Operator Spaces 69

In Chapter 4, we develop the theory of symmetric functionals on symmetric operator spaces. We are especially interested in those functionals which are monotone with respect to Hardy–Littlewood(–Polya) submajorization.

Deﬁnition 2.7.3. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace .LetE.M, / be a symmetric operator space. The functional ' 2 E.M, / is called (a) symmetric if '.A/ D '.B/ whenever .A/ D .B/ for 0 A, B 2 E.M, /.

(b) fully symmetric if '.B/ '.A/ whenever .A/ .B/ for 0 A, B 2 E.M, /. Observe that we do not request in Deﬁnition 2.7.3 (a) that ' is necessarily a positive functional on E.M, /. However, Deﬁnition 2.7.3 (b) requires that fully symmetric functionals be positive since 0 A if 0 A. The Dixmier traces introduced in equation (1.4) in Section 1.3.2, by their construction, are examples of fully symmetric singular traces. The normal trace is a fully symmetric trace on L1.M, /. The following lemma shows the relevance of symmetric functionals to trace theory.

Lemma 2.7.4. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace .LetE.M, / be a symmetric operator space. (a) Every symmetric functional on E.M, / is a continuous trace.

(b) If M is atomless (or atomic) and a factor, then every continuous trace on E.M, / is a symmetric functional.

Proof. The ﬁrst assertion is rather trivial, but the second is not. (a) If 0 A 2 E.M, /and if U 2 M is unitary, then .U AU / D .A/.Since' is symmetric, it follows that '.U AU / D '.A/. By linearity, ' is a trace.

(b) If 0 A, B 2 E.M, / are such that .A/ D .B/, then there exist 0 An, Bn 2 E.M, / such that An ! A, Bn ! B andsuchthat.An/ D .Bn/ are countably-valued functions. By construction, the operators An and Bn have identical discrete spectrum and their corresponding eigenprojections have the same trace. Since M is a factor, it follows that there exists a partial isometry Un which conjugates the corresponding eigenprojections of An and Bn. Conse- quently, An D UBnU and, therefore, '.An/ D '.Bn/.Since' is continuous, it follows that '.A/ D '.B/. As a result, symmetric functionals express the conceptual boundary of continuous traces that may be constructed from singular values. We shall show in Chapter 4 that the construction formula always takes the form of a symmetric functional on a symmetric function space. In this way symmetric functionals express the most general 70 Chapter 2 Preliminaries on Symmetric Operator Spaces possible scope of the Dixmier-style construction of traces. In particular, Lemma 2.7.4 says that, for the von Neumann algebra M D L.H /, every continuous trace on every symmetrically normed ideal of compact operators is a symmetric functional. That M is a factor in Lemma 2.7.4 (b) is the natural limitation on the bijective association between symmetric functionals and traces, as was the case for the Calkin correspondence between Calkin function spaces and operator bimodules. In the disin- tegration of a general von Neumann algebra into factors, the center (where the concept of trace is moot anyway) interrupts the bijective association, as the next example makes clear.

Example 2.7.5. If M D l1, then every continuous linear functional on l1 is a trace. However, there are no symmetric functionals on l1. Continuous traces and symmetrical functionals can be further reﬁned into normal and singular traces and functionals.

Deﬁnition 2.7.6. Let .E.M, /, kkE / be a symmetric operator space. A continuous trace ' 2 E.M, / is called E M E M (a) normal if ' 2 . , /n, that is if X˛ # 0 2 . , / then '.X˛/ ! 0. E M E M (b) singular if ' 2 . , /s , that is if ' vanishes on . , /0. Normal and singular symmetric functionals are similarly deﬁned.

Since L1.M, / D L1.M, /0 is its own separable part, then the noncommutative L1-space admits the normal continuous trace , but does not admit any singular continuous trace. This does not represent the universal case however. Many symmetric operator spaces admit continuous singular traces. As mentioned, a study of continuous singular traces on these symmetric operator spaces leads to new possibilities for noncommutative integration theory. Next, in Part II, we consider if and only if conditions for a symmetric operator space to admit a continuous singular trace, and the form of those traces. We will show that continuous traces decompose uniquely into normal and singular parts, providing a trace version, for all symmetric operator spaces, of the Yosida–Hewett decomposition. We study fully symmetric traces on Lorentz operator ideals in detail in Part III. Unless stated explicitly, every trace we consider on a symmetric operator space is assumed to be continuous.

2.8 Notes

Von Neumann Algebras and Semiﬁnite Normal Traces The uniform, strong and weak operator topologies, and rings of bounded operators closed in those topologies, were introduced by F. Murray and J. von Neumann [166]. Along with C - algebras, von Neumann algebras form one of the pillars of operator algebra theory [63, 243]. Section 2.8 Notes 71

The cited monographs, amongst many others, deal comprehensively with the theory of von Neumann algebras and semiﬁnite normal traces. Theorem 2.1.3 is proved in [217].

Singular Values in Semifinite von Neumann Algebras Singular values for semifinite von Neumann algebras can be traced to the fundamental paper by F. Murray and J. von Neumann [166] and to A. Grothendieck’s announcement[102] (in the special case of finite atomless factors). The general case was almost simultaneously considered by M. Sonis [223] and V. I. Ovchinnikov[174,175]. Later it was studied by F. J. Yeadon [263], T. Fack [88], and T. Fack and H. Kosaki in [90]. A presentation of the subject is given in the forthcoming book [68]. Another frequently used notation for the set S.M, / of -measurable operators is Mf. We refer the reader to [168] for the proof of Lemma 2.3.4 and Theorem 2.3.5. The algebra S.M, /, when equipped with the (so-called) measure topology, is a complete topological algebra and it forms the basis of the theory of noncommutative integration, playing the same role that the algebra S.0, 1/, equipped with the Lebesgue measure, plays in classical integration theory. See [68,90,168,174,175]. The equation (2.2) is a noncommutative analog of the classical formula linking the distribution function of a random variable and its decreasing rearrangement [10,139]. The proof of the noncommutativeversion can be found in [174, Theorem 1], [88, Proposition 1.3], [90, Proposi- tion 2.2] and [68]. The formula given in Example 2.3.9 is classical and still holds for rearrange- ments of measurable functions defined on an arbitrary -finite measure space [10,68,139]. The proof of Lemma 2.3.12 can be found in [68, 88, 90, 174, 263]. The proof of Theo- rem 2.3.13 maybe found in [174, Theorem 1], [90, Proposition 2.4]. The result of Lemma 2.3.15 is well known (see [3] and [39]) and is frequently used.

Calkin Correspondence in the Semiﬁnite Setting The Calkin correspondence for general factors is folklore. D. Guido and T. Isola in [103, Sec- tion 3] provide a similar statement.

Symmetric Operator Spaces Question 2.5.4 naturally arises from J. von Neumann’s paper [172] published in 1937. In this paper von Neumann introduced symmetric norms in the finite-dimensional setting, in particular, for the matrix ideals Lp, five years after Banach’s fundamental book [8] introduced and studied in depth the lp-sequence and Lp-function spaces. In contrast to Banach’s book, the von Neumann paper [172] is almost completely unknown, even to experts. It appeared in an obscure Russian journal, which ceased to exist almost immediately after its first volume was published (the coming Second World War and disruption of scientific contacts did not help either). However, from the present point of view the theory of symmetric operator spaces began with this paper. In fact, von Neumann anticipated the Calkin correspondence by associating with every n-dimensional symmetric sequencespace a n2-dimensionalsymmetric matrix space (coinciding as a linear space with the set of all n n matrices). This connection inspired the theory of symmetrically normed ideals of compact operators, developed by von Neumann with R. Schatten [205–208], and later by Gohberg and Krein [98,99]. The term “symmetrically normed ideal of compact operators” is due to the highly influential monographs [98, 99]. B. Simon in his recommended book [222] prefers a somewhat vaguer term “trace ideals”. Sometimes, there is a difference in the terminology. For example, 72 Chapter 2 Preliminaries on Symmetric Operator Spaces in Theorem 1.16 of [222] the assertion .b/ does not hold for the norm of an arbitrary symmetrically normed ideal (see e.g. corresponding counterexamples in [129, p. 83]). The Question 2.5.4, which is the infinite-dimensional version of von Neumann’s initial matrix result, was fully answered in [129]. An improvedversion of this result is given in Chapter 3. For a thorough survey describing the modern state of this theory we refer to [185]. The general theory of symmetric operator spaces is studied in numerous papers, among which we cite just a few [41, 69–71, 230–234, 263, 264] as the most relevant to our present exposition. For proofs of the Banach space structure of the noncommutative Lp-spaces we refer to [61,105,140,168,217]. The -compact operators in the semifinite setting was considered by M. Sonis [223] and V. I. Ovchinnikov [176]. A different notion of compact operators in the semifinite setting originated with M. Breuer [21,22]. For a beautiful exposition of the theory of symmetric function spaces and its application in interpolationtheory, we refer to [139] and [10]. These books contain a concise exposition of the theory of the Lorentz function spaces M and M , as well as rich historical information and referenceson earlier works in this area. In particular, on the contribution of G. G. Lorentz [156], which led to the introduction of these spaces. These works undoubtedly influenced researchers studying noncommutativeanalogs of these spaces. We observe that in many sources the spaces M are called Marcinkiewicz spaces (see e.g. [139] and many of our own papers [33–35,153, 241]). In this book, however, we follow the terminology of [10] and refer to those spaces as Lorentz spaces. The first example of a symmetric operator ideal which is not fully symmetric is due to G. Russu [200]. It is much harder to find a symmetric operator ideal which is not strongly symmetric. In the setting of symmetric function spaces on .0, 1/ such an example is known and due to A. Sedaev [211], who answered a commutative variant of this question (asked by one of the authors in 1985). Sedaev’s construction is technical and heavily depends on some remarkable facts from the theory of Lorentz function spaces contained in the paper [20] of M. Sh. Braverman and A. A. Mekler. A noncommutative version of Sedaev’s result yielding an example of a symmetric operator space on a II1-factor which does not admit a strongly symmetric norm was presented in [233]. Fully symmetric ideals of compactoperators are Banach ideals of compact operators that are solid under Hardy–Littlewood(–Polya)submajorization . Two-sided ideals of compact operators that are solid under the submajorization (with no Banach space structure assumed) have been studied under the term arithmetically mean closed ideals, see [80,118,121].

Trace Duality, Normal and Singular Functionals In the framework of the theory of Banach lattices, it is customary to consider a decomposition of an arbitrary continuous functional on a given Banach lattice into a direct sum of “normal” (that is continuous with respect to order convergence) and “singular” parts. This decomposition is usually linked with the classical theorem of K. Yosida and E. Hewitt [265], which states that any bounded additive measure can be uniquely represented as the sum of a countably additive measure and a purely ﬁnitely additive measure, the so-called singular part, which is characterized by the fact that its absolute value does not dominate any non-zero positive countably additive measure. It is interesting to observe that this purely commutative result has a noncommutative precursor. Indeed, a noncommutative analog of this result was obtained by J. Dixmier [60]. Dixmier mentions that some of his results had been obtained ear- Section 2.8 Notes 73

lier in papers of R. Schatten and/or R. Schatten and J. von Neumann [205, 207, 208], where it was shown that, using trace duality, the Banach dual of the space of compact operators in a Hilbert space is the trace class operators, and the Banach dual of the trace class operators is the space of all bounded operators. Dixmier’s result, which is based on this duality, states that each bounded linear functional f on the algebra L.H / of all bounded linear operators on a separable Hilbert space is a direct sum of a trace functional g and a singular functional h, vanishing on the compact operators, such that kf kDkgkCkhk. A direct analog of this result for a general von Neumann algebra M was ﬁrst established by M. Takesaki [242]. Takesaki’s result yields that for anyLW -algebra M the following decomposition of the conjugate space M M M M? holds: D , where the direct sum is taken in the l1-sense. In the framework of the approach above, it is natural to study further the singular component of the decomposition. In the Banach lattices setting, the papers by G. Ya. Lozanovskii [157– 159] studied the “singular” part .M /s of the dual to the commutative analogs M of Lorentz operator ideals. This work did not directly lead to the notion of “singular traces” but was nevertheless helpful to recognize the connections between commutative and noncommutative theories (see [67]). For symmetricallynormed ideals of compactoperators a complete analogyto the Yosida and Hewitt result was achievedin the paper [73]. Proposition 2.7 in [73] extendedDixmier’s earlier results on L.H /. For extensions to (fully and strongly) symmetric operator spaces on arbitrary von Neumann algebras we refer the reader to [72] and [64]. It is interesting to observe that the assertion of Theorem 2.6.15 falls short of the l1-type decomposition achieved in [242,265] and [73]. It follows from the proof of [64, Proposition 5.5] (see also comments following Theorem 5.9 in that paper) that we have for the decomposition (2.6)

k'nkM , k'skM 4k'kM .

Part II General Theory

This part of the book discusses the general theory of symmetric functionals on symmetric operators spaces. Chapter 3 confirms that the Calkin correspondence, for atomless or atomic semifinite von Neumann algebras, is a bijective functor between symmetric operator spaces and symmetric function spaces. The technique used is a recent development, and solves a question about symmetrically normed ideals of compact operators which goes back to works of J. von Neumann and R. Schatten. That the Calkin correspondence is a functor of Banach spaces is the necessary prerequisite for discussing the correspondence between symmetric functionals on symmetric operator spaces (which we recall are continuous linear functionals) and symmetric functionals on symmetric function spaces. Chapter 4 confirms that the association between symmetric functionals on operator spaces and symmetric functionals on function spaces is given by the Calkin correspondence. In summary, the situation shown in Chapter 3 and Chapter 4 is the following.

(a) If .E, kkE / is a symmetric function space then the assignment

E.M, / :DfA 2 S.M, / : .A/ 2 Eg

with the norm

kAkE :Dk.A/kE

is a symmetric operator space for a von Neumann algebra M equipped with a ﬁxed faithful normal semiﬁnite trace . Moreover, every symmetric functional f 2 E lifts to a symmetric functional ' 2 E.M, / according to the formula

'.A/ D f..A//, A 0.

(b) Conversely, if M is atomless (or atomic) and .E.M, /, kkE / is a symmetric operator space then the assignment

E :Dfx 2 S : .x/ D .A/ for some A 2 E.M, /g

(here S is the function space in Example 2.3.3 of Chapter 2) with the norm

kxkE :DkAkE 76 Part II General Theory

is a symmetric function (or sequence) space. Moreover, for every symmetric functional ' 2 E.M, / there exists a symmetric functional f 2 E such that

'.A/ D f..A//, A 0.

Chapter 4 continues by addressing the implications of this result for continuous traces on symmetric operator spaces (recall that every symmetric functional is a continuous trace on E.M, /). Specifically, existence of symmetric functionals, existence of fully symmetric functionals, existence of symmetric functionals that are not fully symmetric, and the question of whether the functionals are normal or singular, unique or a plethora, are answered from a study of symmetric functionals on function spaces. The main results are Theorem 4.9.2 and Theorem 4.10.1. As an example of the use of Theorem 4.9.2 and Theorem 4.10.1 we can answer the basic existence questions of continuous traces on Banach operator bimodules of semifinite atomless (or atomic) factors (recall from Lemma 2.4.6 that the symmetric operator spaces and the Banach bimodules of a factor are in bijective correspondence, and from Lemma 2.7.4 that the set of symmetric functionals on a symmetric operator space of a factor is identical to the set of all continuous traces). If M is an atomless (or atomic) semifinite factor and E.M, / is a symmetric operator space equipped with a Fatou norm, then one of the following mutually exclusive possibilities hold. (a) The space E.M, / does not admit a nontrivial continuous trace. (b) The space E.M, /admits a (up to a constant factor) uniquenontrivial continuous trace and it is the (extension of the) faithful normal semifinite trace . (c) The space E.M, / admits an infinite number of nontrivial continuous singular traces (in fact, the set of Hermitian continuous traces is an infinite dimensional Banach lattice).

As an example of (b), see the extension of to the noncommutative L1-space L1.M, / in Example 2.6.2. Indeed, if E.M, / 6 L1.M, / is not a subset of the noncommutative L1-space and is inﬁnite then (b) is impossible, and the case (a) or (c) can be determined by the result that (a) does not hold if there exists a positive 1 ˚ E M m E operator A 2 . , / such that limm!1 m kA k > 0. For the factor M D L.H /, in existing terminology, Chapter 3 and Chapter 4 establish that (a) there is no ambiguity in using the notation for the Calkin correspondence used in Chapter 1 (as a functor between sets) for the Calkin correspondence

: EC ! EC

as a functor between Banach spaces. Part II General Theory 77

(b) any continuous trace ' on a symmetric ideal of compact operators E is obtained from a symmetric functional f on the corresponding Calkin sequence space E according to the formula

'.A/ D f ı .A/, A 0. (II.1)

Formula (II.1) reduces the study of continuous traces on symmetric ideals of compact operators to the study of symmetric functionals on sequence spaces. It raises two further questions for the theory of continuous traces on symmetric ideals of compact operators, which are addressed in Chapter 5 and Chapter 6. In Chapter 5 we show that the Lidskii formula holds for all continuous traces. If E is a symmetric ideal of compact operators and E is the corresponding Calkin sequence space, then an eigenvalue sequence .A/ 2 E for any A 2 E,or

: E ! E, and the Lidskii formula holds for every continuous trace ' on E,

'.A/ D f ı .A/.

In this case the correspondence between continuous traces and symmetric functionals is unequivocal: ' D f ı , f D ' ı diag.

Here diag is the diagonal operator for any orthonormal basis of H from Chapter 1. These result are obtained by the commutator subspace method of Kalton, Kalton and Dykema, and Dykema, Figiel, Weiss and Wodzicki. We also introduce the closed commutator subspace, which is an essential concept for the study of continuous traces (we discuss this in more detail in Chapter 5 and its notes). The second question raised by (II.1) is exactly what do symmetric functionals on symmetric sequence spaces look like? It is nice to have an existence result, but what concrete formulas of singular values give rise to continuous traces? We can answer this question for fully symmetric functionals. Dixmier traces, introduced in Section 1.3.2 of Chapter 1, are an example of a concrete construction. Chapter 6 looks initially at Lorentz operator ideals of L.H /,which fall into the category of fully symmetric operator spaces with Fatou norms, and introduces Dixmier’s construction on a general Lorentz ideal of compact operators. For a Lorentz ideal M , the criteria on existence of symmetric functionals translate to the condition .2t/ lim inf D 1. t!1 .t/ If the condition is satisfied, the Lorentz ideal possesses an infinite number of continuous traces, otherwise none. The first interesting result is that the set of normalized 78 Part II General Theory fully symmetric functionals on these Lorentz ideals and the set of Dixmier traces on them are identical. The fully symmetric functionals are therefore fully constructible. Even more surprisingly, we show that Dixmier traces can be defined on an arbitrary fully symmetric ideal of the algebra L.H /, and that any fully symmetric functional on a fully symmetric ideal is approximated in the weak topology by Dixmier traces. Thus, all fully symmetric functionals are asymptotically constructible according to, basically, Dixmier’s original construction. The construction in Chapter 6 extends some trace results of Wodzicki, but we discuss that in more detail in that chapter and its notes. Chapter 3 Symmetric Operator Spaces

3.1 Introduction

In this chapter we introduce uniform submajorization, which extends the classical notion of Hardy–Littlewood(–Polya) submajorization used extensively in the theory of symmetric function spaces. Uniform submajorization proves an indispensable tool for showing the main results of this chapter and the next, and is one of the central technical devices in symmetric operator space theory. Specifically, the main result of this chapter is the affirmative answer to the Ques- tions 2.5.4 and 2.5.5 posed in Section 2.5 of Chapter 2. We know already from Theo- rem 2.5.3, that if E.M, / is a symmetric operator space for an atomless (or atomic) semifinite von Neumann algebra M then the assignment

E :Dfx 2 S : .x/ D .A/ for some A 2 E.M, /g and

kxkE :DkAkE for all x 2 E, A 2 E.M, / such that .A/ D .x/ deﬁnes a symmetric function (or sequence) space .E, kkE /. This is the relatively easy direction for the Calkin correspondence as a functor between symmetric operator spaces and symmetric function spaces. The converse, Question 2.5.4, represents the more difﬁcult direction. The theorem below was recently proved for M D L.H / in [129] using uniform submajorization, and we provide a systematic, and extended, account of the proof in the following sections.

Theorem 3.1.1. Let M be an atomless (or atomic) von Neumann algebra equipped with a faithful normal semiﬁnite trace . (a) If M is atomless and if .1/ D1, then

E :Dfx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g , kxkE :DkAkE and

E.M, / :DfA 2 S.M, / : .A/ 2 Eg , kAkE :Dk.A/kE

is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric function space on .0, 1/ and a symmetric operator space of .M, /. 80 Chapter 3 Symmetric Operator Spaces

(b) If M is atomless and if .1/ D 1, then

E :Dfx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g , kxkE :DkAkE

and

E.M, / :DfA 2 S.M, / : .A/ 2 Eg , kAkE :Dk.A/kE

is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric function space on .0, 1/ and a symmetric operator space of .M, /.

E :Dfx 2 l1 : .x/ D .A/ for some A 2 E.M, /g , kxkE :DkAkE

and

E.M, / :DfA 2 S.M, / : .A/ 2 Eg , kAkE :Dk.A/kE

is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric sequence space and a symmetric operator space of .M, /. Below M is an arbitrary von Neumann algebra (not necessarily atomless or atomic) equipped with a faithful normal semiﬁnite trace . Question 2.5.5 is answered by:

Theorem 3.1.2. For every symmetric function space E on the semi-axis .0, 1/,the set E.M, / :DfA 2 S.M, / : .A/ 2 Eg equipped with the norm

kAkE :Dk.A/kE is a symmetric operator space of .M, /. The proofs of the theorems stated above are given in Section 3.5 and Section 3.6. In Section 3.6 a variant of Theorem 3.1.2 is given for symmetric sequence spaces.

3.2 Submajorization in the Finite-dimensional Setting

Uniform submajorization differs from its Hardy–Littlewood counterpart only in the infinite dimensional setting. In this section we demonstrate the key ideas in the setting of finitely supported sequences. If x 2 Rn, then (the range of) the singular value function .x/ introduced in Sec- Rn tion 2.3 is a decreasing rearrangement of the vector jxj.Fixx 2 C and define the Section 3.2 Submajorization in the Finite-dimensional Setting 81 set Rn C.x/ :Dfy 2 C : y xg of all positive sequences in Rn submajorized by x. The following lemma is well known, but we provide a short proof.

Rn Lemma 3.2.1. For every x 2 C, the set C.x/ is a convex polyhedron.

Proof. The convexity of C.x/ follows from Lemma 3.3.3. Let Sn be the set of all permutations of the set f0, 1, :::, n 1g, For every permutation 2 Sn, consider the convex polyhedrons Rn A :Dfy 2 C : y..0// y..1// y..n 1//g, whichisanintersectionof2n 1 half-planes, and

Xk Xk n B :Dfy 2 RC : y..l// .l, x/,0 k n 1g, lD0 lD0 whichisanintersectionof2n half-planes. We have [ \ C.x/ D A B . 2Sn

Every set A \ B , 2 Sn, is a convex polyhedron. Hence, C.x/ is a union of at most nŠ convex polyhedrons. However, C.x/ is convex and, therefore, connected. It follows that C.x/ is a polyhedron.

Every polyhedron has a ﬁnite number of extreme points. The following theorem describing the extreme points of the polyhedron C.x/ is folklore.

n Theorem 3.2.2. The only extreme points of C.x/ are those y 2 R with .y/ D .x/Œ0,m for some m

Proof. We prove the assertion using induction on n.Lety 2 C.x/ be an extreme point. Without loss of generality, y D .y/ and x D .x/. Setting m to be the largest value such that .m, y/ > 0, we have y 2 C.xŒ0,m/. Thus, y is an extreme point of the set C.xŒ0,m/.Ifm 0for0 l

Xk min ..l, x/ .l, y// D 0. (3.1) 0k

There exists k

Xk :D minfy.1/,min ..l, x/ .l, y//g > 0. 0k

Xn1 Xn1 Xn1 Xn1 .l, yC/ D .l, y/ D y.l/ .l, x/. lD0 lD0 lD0 lD0

It follows that yC, y 2 C.x/.Sincey D .yC C y/=2, it follows that y is not an extreme point of C.x/. Further, we need a classical result due to C. Caratheodory and E. Steinitz (see [27] and [225]).

Theorem 3.2.3. Let P Rn be a convex polyhedron. Every x 2 P is a convex combination of at most n C 1 extreme points of P .

Proof. By deﬁnition, there exists a sequence xk,0 k m, of extreme points of P and a sequence of positive numbers k,0 k m, such that Xm Xm x D kxk,1D k. kD0 kD0

If m n, then the assertion is proved. Otherwise, the vectors x1 x0, :::, xm x0 2 n R are linearly dependent. Select a non-zero sequence k,1 k m, such that Xm k.xk x0/ D 0. kD1 Section 3.3 Hardy–Littlewood(–Polya) Submajorization 83

P n Setting 1 :D kD2 k,wehave Xm Xm kxk D 0, k D 0. kD0 kD0

There exists a strictly positive number amongst the numbers k,0 k m.Set k ˛ :D inf : k > 0 . k

Since k 0, 0 k m, it follows that k ˛k 0for0 k m and Xm Xm x D .k ˛k/xk, .k ˛k/ D 1. kD0 kD0 At least one of the coefﬁcients in the above sum is 0. Rearranging the summands, if necessary, we obtain mX1 mX1 0 0 x D kxk, k D 1 kD1 kD0

0 with positive numbers k,0 k m 1. Repeat the argument until m does not exceed n.

We are ready for the key result of this section, which describes the set of positive sequences in Rn submajorized by a ﬁxed positive sequence x 2 Rn.

Rn Corollary 3.2.4. If x 2 C, then every y 2 C.x/ is a convex combination of at most .n C 1/ sequences dominated by x. That is, Xn y D kzk, .zk/ .x/,0 k n. kD0 Proof. The assertion follows from Theorem 3.2.3 and Theorem 3.2.2.

3.3 Hardy–Littlewood(–Polya) Submajorization

We introduce submajorization in the sense of Hardy, Littlewood and Polya and we refer an interested reader to their classic book [109]. Another much more modern and very influential book by A. W. Marshall and I. Olkin [161] describes the fundamental value of the submajorization theory and its influence which permeates almost every branch of modern mathematics. We especially emphasize the importance of this theory in the interpolation theory of linear operators [10, 139] and, our interest, the related theory of symmetric function spaces. We repeat Definition 2.5.6. 84 Chapter 3 Symmetric Operator Spaces

Definition 3.3.1. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1/.M, /. The operator B is said to be submajorized by A and written B A if Z Z t t .s, B/ds .s, A/ds, t 0. 0 0 The connection of this definition with classical submajorization theory (that is, for functions or sequences) is immediate: we have

B A , .B/ .A/.

The function Z t t ! .s, A/ds, A 2 S.M, /, t>0 0 is paramount to our study of symmetric spaces and symmetric functionals. The following lemma is an important identiﬁcation.

Lemma 3.3.2. Let M be an atomless semiﬁnite von Neumann algebra. For every t 2 RC and every A 2 .L1 C L1/.M, /, we have Z t .s, A/ds D supf.pjAjp/ : p 2 Proj.M/, .p/ tg. 0

The same assertion is valid for atomic algebras provided that t 2 ZC.

Proof. It follows from Lemma 2.6.3 that Z 1 .pjAjp/ D .s, pjAjp/ds. 0 Note that we have directly from the deﬁnitions

.s, pjAjp/ D 0, 8s>.p/D t and .s, pjAjp/ .s, jAj/, 8s .p/ D t.

Therefore, Z t .pjAjp/ .s, jAj/ds. 0 Now, we prove the converse inequality. Fix >0 and select s 0 such that M njAj.s/ t njAj..1 /s/. Select an arbitrary projection p 2 such that

EjAj.s, 1/ p EjAj..1 /s, 1/, .p/ D t. Section 3.3 Hardy–Littlewood(–Polya) Submajorization 85

Observe that Z t .pjAjp/ .1 / .s, A/ds. 0 Since >0 is arbitrarily small, the assertion follows.

The following two results are fundamental to submajorization theory.

Theorem 3.3.3. Let M be a semiﬁnite von Neumann algebra and let A, B 2 .L1 C L1/.M, /. We have .A C B/ .A/ C .B/.

Proof. By Lemma 2.3.15, there exist partial isometries U , V such that

jA C BjU jAjU C V jBjV .

It is, therefore, sufﬁcient to prove the assertion for the case when both A and B are positive. By Lemma 2.3.18, we may assume without loss of generality that M is atomless. Fix t>0and>0. By Lemma 3.3.2, there exists a projection p 2 M such that .p/ t and such that Z t .s, A C B/ds .p.AC B/p/ C . 0 By Lemma 3.3.2, we also have Z Z t t .pAp/ .s, A/ds, .pBp/ .s, B/ds. 0 0 Thus, Z Z Z t t t .s, A C B/ds .s, A/ds C .s, B/ds C . 0 0 0 Since is arbitrarily small, the assertion follows.

The next result is a companion to Theorem 3.3.3. For s>0, the dilation operator s acting on the space L0.0, 1/ of all measurable functions is deﬁned by setting t . x/.t/ :D x , x 2 L .0, 1/, t>0. s s 0

Theorem 3.3.4. Let M be a semiﬁnite von Neumann algebra and let A, B 2 .L1 C L1/.M, / be positive operators. We have

.A/ C .B/ 21=2.A C B/. 86 Chapter 3 Symmetric Operator Spaces

Proof. By Lemma 2.3.18, we may assume without loss of generality that M is atomless. Fix t>0and>0. By Lemma 3.3.2, there exist projections p, q 2 M such that .p/, .q/ t and Z Z t t .s, A/ds .pAp/C , .s, B/ds .qBq/C . 0 0 Define a projection r D p _ q. It follows that .r/ .p/C .q/ D 2t. Hence, Z 2t .pAp/ C .qBq/ .r.AC B/r/ .s, A C B/ds. 0 Therefore, Z Z Z t t 2t .s, A/ds C .s, B/ds .s, A C B/ds C 2. 0 0 0 Since >0 is arbitrarily small, the assertion follows. We also need a slight generalization of the preceding assertion. This result provides a fundamental estimate, which will be connected to uniform submajorization in subsequent sections. M L Lemma 3.3.5. Let be a semifinite von Neumann algebra. Let Ak 2 . 1 C L M N R N P1/. , /, k 2 , be positive operators and let ˛k 2 C, k 2 , be such that 1 kD1 ˛k 1. We have Z Z X1 ˛ka a X1 .s, Ak /ds s, Ak ds, 8a>0. 0 0 kD1 kD1 P 1 L L M Here, we assume that the series kD1 Ak converges in . 1 C 1/. , /. Proof. By Lemma 2.3.18, we may assume without loss of generality that M is atom- N less. Fix >0. For every k 2 , we can find a projection pk such that .pk/ D ˛ka and Z ˛ka k .s, Ak / 2 C .pk Akpk/. 0 P 1 1 Set p D_kD1pk.Since kD1 ˛k 1, it follows that .p/ a. It follows that Z X1 ˛ka X1 X1 .s, Ak / C .pkAk pk/ C .pAk p/ 0 kD1 kD1 kD1 Z X1 a X1 D C p Ak p C s, Ak ds. 0 kD1 kD1 Since >0 is arbitrarily small, the assertion follows immediately. Section 3.3 Hardy–Littlewood(–Polya) Submajorization 87

The following characterization of Hardy–Littlewood submajorization is known for the classical setting M D L.H / (see [98, Chapter II, Lemma 3.4]). We present a short proof of the general result.

Theorem 3.3.6. If 0 A, B 2 .L1 C L1/.M, /, then B A if and only if

..B t/C/ ..A t/C/, 8 t>0. (3.2)

Proof. Fix t>0. We have ..A t/C/ D ..A/ t/C. Applying Lemma 2.6.3 to the operator .A t/C, we obtain Z Z 1 nA.t/ ..A t/C/ D ..s, A/ t/Cds D ..s, A/ t/ds. (3.3) 0 0 Computing the derivative of the function Z u u ! ..s, A/ t/ds 0 we see that this function attains its maximum at u D nA.t/. If B A,then Z Z Z nB .t/ nB .t/ nA.t/ ..s, B/ t/ds ..s, A/ t/ds ..s, A/ t/ds. 0 0 0 Inequality (3.2) now follows from (3.3). Suppose now that (3.2) holds. Fix u>0andsett D .u, A/. It follows that Z Z u nB .t/ ..s, B/ t/ds ..s, B/ t/ds D ..B t/C/ 0 0 Z u ..A t/C/ D ..s, A/ t/ds. 0

Hence, Z Z u u .s, B/ds .s, A/ds. 0 0 Since u is arbitrary, we have B A. We complete this section with a lemma, which provides a useful submajorization estimate for the direct sum introduced in Deﬁnition 2.4.3.

Lemma 3.3.7. Let M be a semiﬁnite von Neumann algebra and let Ak 2 .L1 C L1/.M, /, k 0, be positive operators. It follows that Z Z t X1 t M1 s, Ak ds s, Ak ds. 0 0 kD0 kD0 88 Chapter 3 Symmetric Operator Spaces

Proof. Let M0 be the commutative von Neumann algebra constructed in Theo- S M rem 2.3.11. Choose positive operators Bk 2 . 0, /, k 0, such that

.Bk/ D .Ak/, Bk .1, Bk/EBk .0, 1/, k 0, andsuchthatBkBl D 0fork ¤ l. For a given t>0, there exists a projection p 2 M0 such that .p/ D t and Z t X1 X1 s, Bk ds D p Bk p . 0 kD0 kD0 M Deﬁne pairwise orthogonalP projections pk :D p ^ EBk .0, 1/.Since 0 is commu- 1 tative, it follows that p D kD0 pk. Observe that pkBl D pl Bk D 0fork ¤ l and, therefore, Z Z t X1 X1 X1 .pk/ s, Bk ds D pkBkpk s, Ak ds. 0 0 kD0 kD0 kD0 It now follows from Corollary 3.3.5 that P Z 1 Z 1 t X k0 .pk/ X s, Bk ds s, Ak ds. 0 0 kD0 kD0 P Since k0 .pk/ D t, the assertion follows.

The next section discusses uniform submajorization, which is an extension to the Hardy–Littlewood submajorization theory.

3.4 Uniform Submajorization

The following deﬁnition, introduced originally in [129], plays a major role in the Calkin correspondence and in our treatment of singular traces.

Deﬁnition 3.4.1. Let M be a semiﬁnite von Neumann algebra and let A, B 2 .L1 C L1/.M, /. We say that B is uniformly submajorized by A (written B C A)ifthere exists 2 N such that Z Z b b .s, B/ds .s, A/ds, a b. (3.4) a a Uniform submajorization is a stronger condition than the Hardy–Littlewood submajorization introduced in the last section. Our main objective in this chapter is The- orem 3.4.2 describing (in the cases M D l1, M D L1.0, 1/ and M D L1.0, 1/) the convex hull of the set f0C : .C / .A/g in terms of uniform submajorization. Section 3.4 Uniform Submajorization 89

Theorem 3.4.2. Let 0 x, y 2 l1. (a) If y belongs to the convex hull of the set f0 z : .z/ .x/g, then y C x.

(b) If y C x, then, for every >0, the element .1 /y belongs to the convex hull of the set f0 z : .z/ .x/g.

The same assertion holds for functions x, y 2 .L1 C L1/.0, 1/ (or x, y 2 L1.0, 1/). The following corollary identiﬁes symmetric sequence (or function) spaces as monotone with respect to uniform submajorization. This identiﬁcation is our main tool in linking commutative and noncommutative symmetric spaces.

Corollary 3.4.3. Let E be a symmetric sequence space and let x 2 E. If y 2 l1 is such that y C x, then y 2 E and kykE kxkE . The same assertion holds for symmetric function spaces.

N Proof. Fix >0. By Theorem 3.4.2 (b), there exist n 2 ,0 zk 2 E,1 k n, and positive numbers k,1 k n, such that .zk/ .x/ for every 1 k n and Xn Xn .1 /y D kzk, k D 1. kD1 kD1

Therefore, Xn Xn .1 /kykE kkzkkE kkxkE DkxkE . kD1 kD1 Since >0 is arbitrarily small, the assertion follows.

Proof of Theorem 3.4.2 We start the proof with a simple lemma which strengthens Theorem 3.3.3 and Theo- rem 3.3.4 by replacing Hardy–Littlewood submajorization with its uniform counterpart.

Lemma 3.4.4. Let M be a semiﬁnite von Neumann algebra and let A, B 2 .L1 C L1/.M, / be positive operators. We have

.A C B/ C .A/ C .B/ C 21=2.A C B/.

Proof. It follows from Theorem 3.3.3 that Z Z b b .s, A C B/ds ..s, A/ C .s, B//ds. 0 0 90 Chapter 3 Symmetric Operator Spaces

It follows from Theorem 3.3.4 that Z Z 2a a .s, A C B/ds ..s, A/ C .s, B//ds. 0 0 Subtracting the inequalities, we obtain Z Z b b .s, A C B/ds ..s, A/ C .s, B//ds. (3.5) 2a a This proves the ﬁrst inequality. The proof of the second inequality is identical.

For all positive x, y 2 l1,weset XN ŒŒy, x :D inf N : y xj , .xj / .x/, xj 0 . j D1

We employ the same notation for positive functions x, y 2 .L1 C L1/.0, 1/ (or x, y 2 L1.0, 1/). We study the properties of ŒŒ, as the proof of Theorem 3.4.2 critically depends on them. The following lemma establishes the convexity of the operation y ! ŒŒy, x for an arbitrary 0 x 2 l1. It also provides a formula for the Minkowski functional of the set

Q.x/ :D the convex hull of the set of sequences f0 z : .z/ .x/g.

We denote the latter functional by

hz; xi :D inff>0: z 2 Q.x/g

(and hz; xi :D1if z is not in the linear span of Q.x/).

Lemma 3.4.5. The mapping .x, y/ ! ŒŒx, y satisﬁes the following properties.

(a) If y1 y2, then ŒŒy1, x ŒŒy2, x. If x1 x2, then ŒŒy, x2 ŒŒy, x1.

(b) For all positive x, y1, y2 2 l1, we have

ŒŒy1 C y2, x ŒŒy1, x C ŒŒy2, x.

(d) For a ﬁxed positive x 2 l1, and every positive y 2 l1, we have 1 hy; xiD lim ŒŒNy, x. (3.6) N !1 N

The same assertion holds for functions x, y 2 .L1 C L1/.0, 1/ (or x, y 2 L1.0, 1/).

Proof. The ﬁrst and second properties follow directly from the deﬁnition. The third property follows from the second one and Lemma 3.4.6, which is a standard Fekete Lemma. In order to prove the fourth property, let x D .x/ and for an arbitrary >0, let N 2 N, cj 0, 1 j N ,andxj 2 l1,1 j N , be such that .xj / D x for 1 j N and XN XN y cj xj , cj < hy; xiC. j D1 j D1

Then, for every positive integer M ,wehave

XN My .ŒMcj C 1/xj j D1 where Œa is the integral part of a. Consequently,

XN ŒŒMy, x .ŒMcj C 1/

Conversely, for any given M 2 N,letK D ŒŒMy, x and let h1, :::, hK 0besuch that .hj / D .x/ D x, j D 1, 2, ..., K and

XK My hj . j D1 Then we have XK 1 K y h Q.x/, M j M j D1

1 and, consequently, hy; xiM ŒŒMy, x. Hence, 1 N hy; xi ŒŒMy, x hy; xiC C . M M 92 Chapter 3 Symmetric Operator Spaces

Letting M !1and keeping in mind that N is ﬁxed and >0 is arbitrary, we obtain the equality (3.6).

Lemma 3.4.6 (Fekete Lemma). If ak, k 1, is a positive sequence such that akCl ak C al for k, l 1, then a a lim n D inf n . n!1 n n0 n

Proof. Fix n 2 N.Wehaveamn man for m 2 N. Thus, for k D mnCs,0 s

Lemma 3.4.7. If xn, yn 2 l1, n 1, are positive sequences, then M1 M1 yn, xn supŒŒyn, xn. n2N nD1 nD1

The same assertion holds for positive functions xn, yn 2 .L1CL1/.0, 1/ (or xn, yn 2 L1.0, 1/), n 1. Proof. Set

Nn D ŒŒyn, xn, N D supŒŒyn, xn D sup Nn. n2N n2N If N D1, then the assertion is trivial. Suppose that N<1. For every n 2 N,there exist xnk 2 l1,1 k Nn, such that .xnk/ .xn/ and

XNn yn xnk. kD1

Setting xnk D xn for Nn

Since M1 M1 xnk xn , nD1 nD1 the assertion follows.

The following lemma is a ﬁnite-dimensional result based on Corollary 3.2.4.

RN N Lemma 3.4.8. Let x, y 2 C be such that y x. For every M 2 , we have

ŒŒMy, x M C N C 1.

Proof.Sincey 2 C.x/, it follows from Corollary 3.2.4 that there exists a sequence xk,0 k N , of extreme points of C.x/ and a sequence of positive numbers k, 0 k N , such that XN XN y D kxk,1D k. kD0 kD0

By Theorem 3.2.2, .xk/ .x/ for every 0 k N . Hence, there exist zk xk such that .zk/ D .x/. It clearly follows that

XN XN My D Mkxk .1 C ŒM k/zk. kD0 kD0

Therefore, XN ŒŒMy, x .1 C ŒM k/ M C N C 1. kD0

The following lemma is an inﬁnite-dimensional surrogate for the preceding result.

Lemma 3.4.9. Let x D .x/ 2 l1 and y D .y/ 2 l1 be such that y xŒu,1/ for some u 2 N. If jsupp.y/jN , then for every M 2 N we have

N ŒŒMy 1 , x M C . Œu, / u

The same assertion holds for functions x, y 2 .L1 C L1/.0, 1/ (or x, y 2 L1.0, 1/).

Proof. Deﬁne sequences , 2 l1 by setting .n/ :D x..n C 1/u/ and .n/ :D y..n C 1/u/ for all n 0. We have

.yŒu,1// .u/ .y/ .xŒu,1// .u/ .x/. 94 Chapter 3 Symmetric Operator Spaces

Hence, u u and, therefore, . Clearly, jsupp./j.N u/=u.By Lemma 3.4.8, N u N ŒŒM, M C C 1 M C . u u

On the other hand, it follows from Lemma 3.4.7 that

N ŒŒMy 1 , x ŒŒM , ŒŒM, M C . Œu, / u u u The next lemma, together with Lemma 3.4.9, provides a crucial technical estimate for Proposition 3.4.11.

Lemma 3.4.10. Let x, y 2 l1 be positive sequences such that y C x. For 2 N as in Deﬁnition 3.4.1 and for every u 0, we have

.y/ 2 .x/ 1 . Œ u,1/ 1 Œu, /

The same assertion holds for functions x, y 2 .L1 C L1/.0, 1/ (or x, y 2 L1.0, 1/).

Proof. For every t>0, we have Z Z Z uCt uCt t C . 1/u uCt .s, y/ds .s, x/ds .s, x/ds. u u t u Let Z Z uCt uCt A D t 0: .s, y/ds .s, x/ds . u 1 u

For every t 2 A,wehavet<. 1/2u.Foreveryt sup.A/,wehave Z Z uCt uCt .s, y/ds .s, x/ds. (3.7) u 1 u Observe that A is a closed set and, therefore, sup.A/ 2 A. Thus, Z Z uCsup.A/ uCsup.A/ .s, y/ds D .s, x/ds. (3.8) u 1 u Subtracting (3.8) from (3.7), we obtain Z Z uCt uCt .s, y/ds .s, x/ds. (3.9) uCsup.A/ 1 uCsup.A/ Section 3.4 Uniform Submajorization 95

It follows from (3.9) and the fact that sup.A/ < . 1/2u that .y/ 2 .y/ C 1 .x/ C 1 Œ u,1/ Œu sup.A/, / 1 Œu sup.A/, / .x/ 1 . 1 Œu, / The following proposition connects uniform submajorization and the Minkowski functional for Q.x/,0 x 2 l1 (whose value is provided in Lemma 3.4.5).

Proposition 3.4.11. If x D .x/ 2 l1 and y D .y/ 2 l1 are such that y C x, then 1 lim ŒŒNy, x 1. N !1 N

The same assertion holds for functions x, y 2 .L1 CL1/.0, 1/ (or, x, y 2 L1.0, 1/). Proof. Let 3 be as in Deﬁnition 3.4.1. For 0 k<,set [1 3k 3kC3C3.n1/ 3kC3n A0k D Œ0, /, Ank D Œ , /, n 1, Ak D Ank nD0 [1 3k 3kC3.n1/ 3kC3n Z B0k D Œ0, /, Bnk D Œ , /, n 1, C D Bnk. nD0

It follows from Lemma 3.4.10 (with u D 23kC3.n1/)that y 3kC2C3.n1/ x 3kC3.n1/ . (3.10) Œ2 ,1/ 1 Œ2 ,1/ Hence, y 3kC2C3.n1/ 3kC3n x 3kC3.n1/ 3kC3n . (3.11) Œ2 , / 1 Œ2 , / The left-hand side of (3.11) has a support of length at most 3kC3n. For every n 1, it follows from (3.11) and Lemma 3.4.9 (with u D 3kC3.n1/)that hh ii 3 My 2 3kC3.n1/ 3kC3n , x 3kC3.n1/ 3kC3n M C . Œ.2 C1/ , / 1 Œ , / Since 3 (in particular, 22 C 1 3), it follows that for n 1, we have hh ii My , x M C 3. (3.12) An,k 1 Bn,k The inequality (3.12) for n D 0 follows from Lemma 3.4.8. 96 Chapter 3 Symmetric Operator Spaces

It follows from (3.12), and Lemma 3.4.7, that hh ii hh 1 1 ii M M My , x D My , x M C 3. (3.13) Ak 1 Ank 1 Bnk nD0 nD0

However, for every m>0 there exists exactly one 0 k<such that m … Ak.It follows that X1

y. 1/ C yŒ0,1/ D yAk . kD0 It follows from (3.13) that

hh ii 1 hh ii X M. 1/y, x My , x .M C 3/. 1 Ak 1 kD0 Let M D 1 C ŒN =. 1/2. It follows that hh ii ŒŒNy, x ŒŒM. 1/21y, x D M. 1/y, x 1 .M C 23/ N. /2 C 23. 1 Letting N !1, we obtain 1 2 lim sup ŒŒNy, x . N !1 N 1 Since can be chosen arbitrary large, the assertion follows. We are now ready to prove Theorem 3.4.2.

Proof of Theorem 3.4.2. Let x, y 2 l1 be positive sequences. Without loss of generality, we assume that x D .x/ and y D .y/.

(a) By assumption, there exist n 2 N, k 0andzk,1 k n, such that .zk/ .x/ and such that Xn Xn kzk D y, k D 1. kD1 kD1 It follows from Lemma 3.4.4 that Xn Xn y C k.zk/ k.x/ D .x/. kD1 kD1 Hence, y C x. Section 3.5 Symmetric Operator Spaces from Symmetric Function Spaces 97

(b) If y C x, then it follows from Proposition 3.4.11 that

1 lim ŒŒNy, x 1. N !1 N By Lemma 3.4.5, the latter limit is the Minkowski functional for the convex hull of the set fz 0: .z/ .x/g. The assertion now follows from the deﬁnition of the Minkowski functional.

If x, y 2 .L1 C L1/.0, 1/ or x, y 2 L1.0, 1/, then the argument remains the same.

3.5 Symmetric Operator Spaces from Symmetric Function Spaces

The main aim of this section is to prove, using Theorem 3.4.2, that the operator space

E.M, / :DfA 2 S.M, / : .A/ 2 Eg associated to a symmetric function space E is a symmetric operator space with norm

kAkE :Dk.A/kE , A 2 E.M, /.

First we show that E.M, / is normed.

Corollary 3.5.1. Let M be a semiﬁnite von Neumann algebra. Let E be a symmetric function space. It follows that the corresponding symmetric operator space E.M, / is a normed space.

Proof.LetA, B 2 E.M, /. We show that the triangle inequality holds for kkE .The other properties are evident. It follows from Lemma 2.3.15 that

jA C BjU AU C V BV for some partial isometries U , V 2 M. It follows from Lemma 3.4.4 that

.A C B/ .U AU C V BV / C .A/ C .B/.

By Corollary 3.4.3,

kA C BkE k.A/kE Ck.B/kE DkAkE CkBkE .

We now show that E.M, / is complete. The proof is based on the fundamental estimate obtained in Lemma 3.5.4. 98 Chapter 3 Symmetric Operator Spaces

Lemma 3.5.2. Let M be a semiﬁnite von Neumann algebra. Let Ak 2 .L1 C L M N R N P1/. , /, k 2 , be positive operators and let ˛k 2 C, k 2 , be such that 1 kD1 ˛k 1. It follows that Z Z b X1 X1 b s, Ak ds .s, Ak /ds,0

Proof. It follows from Lemma 3.3.5 that Z Z a X1 X1 ˛ka s, Ak ds .s, Ak /ds. (3.14) 0 0 kD1 kD1

On the other hand, we have Z Z b X1 X1 b s, Ak ds .s, Ak /ds. (3.15) 0 0 kD1 kD1

Subtracting (3.14) from (3.15), we obtain the assertion.

M L Lemma 3.5.3. Let be a semiﬁnite von Neumann algebra.P Let Ak 2 . 1 C L M N R N 1 1/. , /, k 2 , and let ˛k 2 C, k 2 , be such that kD1 ˛k 1. It follows that Z Z b X1 X1 b s, Ak ds .s, Ak /ds,0

Proof. Fix n 2 N. By Lemma 2.3.15, there exist partial isometries Uk,1 k n, such that ˇ ˇ ˇ Xn ˇ Xn ˇ ˇ Ak Uk jAk jUk. kD1 kD1

It follows from Lemma 3.5.2 that Z Z b Xn Xn b s, Uk jAk jUk ds .s, Uk jAk jUk/ds, 80

Therefore, Z Z b Xn Xn b s, Ak ds .s, Ak /ds, 80

Lemma 3.5.4. Let M be a semiﬁnite von Neumann algebra and let Ak 2 .L1 C L1/.M, /. We have X1 X1 C Ak 2 2k .Ak/. kD1 kD1 P 1 L L M Here, theP series kD1 Ak is assumed to be convergent in . 1 C 1/. , / and the 1 series kD1 2k .Ak / is assumed to be convergent in S (convergent in measure). Proof.Let0< 2a

Set Z k 2kb 2 N k D .s, Ak /ds, 8k 2 . b 2a 21ka

k It is clear that .s, Ak / k for every s 2 b and .s, Ak / k for every s 21ka. Therefore, Z Z Z b 2kb b 1k .s, Ak /ds .s, Ak /ds C kds D .b 2 a/k 21ka 21ka 2kb and Z Z Z b 2kb 21ka k .2k .Ak //.s/ds 2 .s, Ak /ds C kds .b a/k. a 21ka 2ka 100 Chapter 3 Symmetric Operator Spaces

Therefore, Z Z b b 1k .s, Ak /ds .b 2 a/k 2.b a/k 2 .2k .Ak //.s/ds. 21ka a Hence, Z Z b X1 b X1 s, Ak ds 2 2k .Ak / .s/ds. 2a a kD1 kD1

The next theorem is the main result of this chapter. It answers Question 2.5.4 in the afﬁrmative for the atomless case, and it answers Question 2.5.5.

Theorem 3.5.5. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . For every symmetric function space E on .0, 1/ if .1/ D1(or .0, 1/ if .1/ D 1), the set

E.M, / :DfA 2 S.M, / : .A/ 2 Eg is a symmetric operator space when equipped with the norm kAkE :Dk.A/kE , A 2 E.M, /.

Proof. By Corollary 3.5.1, E.M, / is a normed space. We only have to prove completeness. For this purpose, ﬁx a Cauchy sequence fAngn2N in E.M, /. We will prove the existence of A 2 E.M, / such that An ! A in E.M, /. k For every k>0, there exists mk such that kAm Amk kE 4 for m mk. Set B D A A . Clearly, kB kE 4 k for every k 2 N. In partic- k mkC1 mk k P k 1 L ular, kBkkL1CL1 4 and, therefore, the series kD1 Bk converges in . 1 C L1/.M, / by Example 2.6.7. It follows from Lemma 3.5.4 that X1 X1 C Bk 2 2kC1n.Bk /, n 3. kDn kDn

It now follows from Theorem II.4.5 of [139] that X1 X1 X1 kC1n 2n k 32n kC n 2 1 .Bk / 2 2 kBkkE 2 2 D 2 . E kDn kDn kDn

Therefore, X1 2kC1n .Bk/ ! 0 kDn Section 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces 101

P P 1 1 in E. By Corollary 3.4.3, kDn BPk 2 E and . kDn Bk/ P! 0inE. By con- E M 1 E M 1 struction of . , / we have that kDn Bk 2 . , / and kDn Bk ! 0in E.M, /.

The next section gives a similar theorem for the case of a symmetric sequence space.

3.6 Symmetric Function Spaces from Symmetric Sequence Spaces

Theorem 3.5.5 shows that a symmetric function space on the semi-axis defines a symmetric operator space on a semifinite von Neumann algebra M.Inthissectionwe provide the counterpart for a symmetric sequence space. We do so by associating a symmetric function space on .0, 1/ to a symmetric sequence space E. Let A DfAkg be a (finite or infinite) sequence of disjoint sets of finite measure and denote by A the collection of all such sequences. We need the notion of an expectation operator (see [20]).

Deﬁnition 3.6.1. The expectation operator E.jA/ : .L1 C L1/.0, 1/ ! .L1 C L1/.0, 1/ (or E.jA/ : L1.0, 1/ ! L1.0, 1/) is deﬁned by setting X Z E A 1 .xj / :D x.s/ds Ak . m.Ak / Ak k

Lemma 3.6.2. For every x 2 .L1 C L1/.0, 1/ and for every A 2 A, we have E.xjA/ x.

Proof. Every expectation operator is a contraction in both L1 and L1. The assertion now follows from Theorem II.3.4 of [139].

In this section, A Df.n 1, ngn2N is a partition of the semi-axis. Clearly, E.jA/ maps .L1 C L1/.0, 1/ into the set of step functions which can be identiﬁed with sequences.

Construction 3.6.3. Let E be a symmetric sequence space. Let F be the linear space of all x 2 .L1 C L1/.0, 1/ for which E..x/jA/ 2 E. The main aim of this section is to introduce a Banach norm on the function space F so that F becomes a symmetric function space on the semi-axis.

Lemma 3.6.4. Let M be an atomless semiﬁnite von Neumann algebra. If A, B 2 .L1 C L1/.M, / are positive operators, then E A C E A C E A ..A C B/j / ..A/ C .B/j / 21=2 ..A C B/j /. 102 Chapter 3 Symmetric Operator Spaces

Proof. We only prove the left-hand side inequality. Proof of the right-hand side inequality is identical. If b is integer, then it follows from Lemma 3.3.3 that Z Z b b E..A C B/jA/.s/ds D .s, A C B/ds 0 0Z b ..s, A/ C .s, B//ds Z 0 b D E..A/ C .B/jA/.s/ds. 0 However, the inequality Z Z b b E..A C B/jA/.s/ds E..A/ C .B/jA/.s/ds (3.16) 0 0 is piecewise linear and, therefore, holds for every b>0. If 2a is integer, then it follows from Lemma 3.3.5 that Z Z 2a 2a E..A C B/jA/.s/ds D .s, A C B/ds 0 0Z a ..s, A/ C .s, B//ds Z0 a E..A/ C .B/jA/.s/ds. 0 However, the inequality Z Z 2a a E..A C B/jA/.s/ds E..A/ C .B/jA/.s/ds (3.17) 0 0 is piecewise linear and, therefore, holds for every a>0. Subtracting this inequality, we obtain Z Z b b E..A C B/jA/.s/ds E..A/ C .B/jA/.s/ds (3.18) 2a a

Corollary 3.6.5. The function space F given in Construction 3.6.3 admits a norm deﬁned by setting kxkF :DkE..x/jA/kE , x 2 F .

Proof. It follows from Lemma 3.6.4 that

E..x C y/jA/ C E..x/ C .y/jA/ Section 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces 103

provided that x, y 2 .L1 C L1/.0, 1/ are positive functions. By Theorem 3.4.2 and the triangle inequality in E,wehave

kE..x C y/jA/kE kE..x/jA/kE CkE..y/jA/kE .

Theorem 3.6.6. The function space F equipped with the norm in Corollary 3.6.5 is a symmetric function space.

Proof. The norm kkF is already symmetric. We only have to prove the completeness of .F , kkF /. Fix a Cauchy sequence xm 2 F , m 0. By Example 2.6.7, xm, m 0, is a Cauchy sequence in .L1 C L1/.0, 1/ and therefore xm ! x in .L1 C L1/.0, 1/ for some x 2 .L1 C L1/.0, 1/. k For every k 0, there exists mk such that kxm xmk kF 4 for m mk.Set k Pyk D xmkC1 xmk . Clearly, kykkF 4 for every k 0. In particular, the series 1 kDn yk converges to x xmn in .L1 C L1/.0, 1/ for every n 1. E A k We have k ..yk/j /kE 4 . By Theorem II.4.5 of [139], we have E A k k2k ..yk/j /kE 2 . Therefore, X1 X1 E A k 1n k 2k ..yk /j /kE 2 D 2 . (3.19) kDn kDn

On the other hand, we have X1 X1 E A E A 2k .yk /j D 2k ..yk/j /. kDn kDn

It follows from the preceding formula and the deﬁnition of norm in F that

1 1 X X E A 1n 2k .yk/ D 2k ..yk /j / 2 . F E kDn kDn

It follows from Lemma 3.5.4 and Corollary 3.4.3 that X1 X1 2n yk 2 2k .yk/ 2 . (3.20) F F kDn kDn

Thus, the subsequence xmn converges to x in F . Hence, the sequence xn converges to x in F .

We can associate to the symmetric sequence space E (through the symmetric function space F ) a symmetric operator space. 104 Chapter 3 Symmetric Operator Spaces

Corollary 3.6.7. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . For every symmetric sequence space E, the set

E.M, / :DfA 2 S.M, / : E..A/jA/ 2 Eg is a symmetric operator space when equipped with the norm kAkE :DkE..A/jA/kE , A 2 E.M, /.

Proof. We apply Theorem 3.5.5 to the symmetric function space F in Theorem 3.6.6. The assertion follows.

If M is atomic (recall from Section 2.3 that the standing assumption on an atomic semiﬁnite von Neumann algebra is that .p/ D 1 for any non-zero minimal projection p 2 Proj.M/), then E..A/jA/ D .A/ for any A 2 S.M, /. Corollary 3.6.7 answers Question 2.5.4 for the atomic case.

3.7 Notes

Hardy–Littlewood Submajorization The text already mentioned the foundational book of Hardy, Littlewood and Polya [109] and the more modern book by A. W. Marshall and I. Olkin [161]. For applications of submajorization theory to the interpolation theory of linear operators, see [10,139]. Lemma 3.3.7 is a slight generalization of Lemma 2.3 from [40]. The combined result given in Theorems 3.3.3 and 3.3.4 can be found in [103, Proposition 1.10 (i)].

Completeness Solving Question 2.5.4 had been a long standing goal of the second named author. In fact, The- orem 3.1.1 appeared in the paper [129] as a combination of [129, Theorem 8.7] and [129, The- orem 8.11]. These results generalize a number of earlier results in the literature, in particular, where the completeness of symmetric operator spaces was established under certain additional conditions on the (quasi-)norm kkE (see e.g. [262], [264, Proposition 2.8 (ii)], [232, Corol- lary 1], [233, Theorem 1.2.4], [260, Lemma 4.1], [69, Theorem 4.5], [71, Corollary 2.4]). We refer the reader to the forthcoming paper [229] for the afﬁrmative solution for quasi-normed symmetric operator spaces.

Uniform Submajorization Uniform submajorization,as a meansto the solution of Question 2.5.4, was introducedin [129]. Theorem 3.4.2 is a special case of a stronger result obtained in [129]. See also the comments in [183]. For the historical background to the paper [129] we quote from a memorial written by the secondnamed author; reprintedwith permission from the NigelKalton Memorial Website1. In 1998, together with B. de Pagter, P. Dodds and E. Semenov, we approached the notion of a Dixmier trace, which we called, at that time, a symmetric functional, as a part of a general theory of singular traces on symmetrically normed

1 http://kaltonmemorial.missouri.edu/ Section 3.7 Notes 105

operator ideals of L.H / (and, more generally, on symmetrically normed operator bimodules on semifinite von Neumann algebras). At that time, I was vaguely aware of earlier work of Nigel with Figiel and with Dykema. However, their framework was subtly different and I could not immediately see its implications for our own work. [A] meeting in Oberwolfach[in 2004] was a perfect chance to tell to Nigel about our results and also state a number of problems some of which we tried ourselves and some of which were fairly new and which I thought of as important from the viewpoint of noncommutative geometry. Somewhat incredi- bly, Nigel was interested and the very next day he approachedme with a tentative solution to the problem which we thought was very hard. Nigel suggested a way how one can attempt to construct symmetric functionals (and unitarily invariant linear functionals on a special Lorentz ideal of importance in noncommutative geometry) which were not monotone with respect to the Hardy–Littlewood [sub]majorization. The idea was so nice that I liked it straight away. The problem was with its technical implementation, or rather with my understanding of the latter. It was some time before I could present a strict and complete technical record of Nigel’s idea, which was initially stated in two or three lines on a ta- ble napkin. Indeed, many times during my subsequent collaboration with Nigel, I noted that he didn’t seem to experience the technical difficulties which would set back anybody else. Back in 2004, we parted with the agreement that I would try to continue working with the argument and see for what class of Lorentz spaces it is applicable.That took a few years (the paper was only published in 2008) and a few meetings in Adelaide and Missouri which still are (and forever will be) a treasured part of my memory. Our further collaboration was firmly centered on the theory of singular traces. I was able to contribute to our work my knowledge of (sometimes obscure) works on the theory of symmetric spaces from the former Soviet Union (Braverman, Mekler, Russu, Sedaev);however, the earlier ideas and techniquesof Nigel (from his papers with Figiel and Dykema, which I mentioned earlier) have come to play an essential role in our approach. It was an absolute pleasure for me to see how ideas and techniques born from completely different perspectives (and motivations) became central to the study of singular traces and their applications in noncommutative geometry. Eventually, it had become clear to us that there exists a single thread which permeates works on commutators, unitary orbits, various geometrical questions in the theory of symmetric operator spaces and noncommutative geometry which was both fascinating and fruitful. Of course, this realization would never have happened if Nigel was not the mathematical giant that he was. However, our main achievement with Nigel belongs to a different area (even though its applications, by now, have proved paramount for singular trace theory as well). That main achievement is an infinite-dimensional analogue of a finite-dimensional result of John von Neumann from 1937 and the new notion which was invented to obtain this analogue. In that remarkable paper, von Neu- mann laid the foundationof what was later to become the theory of symmetrically normed (or unitarily-invariant) operator ideals. It is also the first paper where the so-called “noncommutative Lp-spaces” made their appearance. I read this pa- 106 Chapter 3 Symmetric Operator Spaces

per of von Neumann as a very young man and realized the beauty of its ideas and noted that it suggested immediately a number of infinite-dimensional questions which were to occupy me for the next 25 years. One of these problems which I tried to resolve in my PhD thesis (1988) was whether a positive unitarily- invariant functional on a given unitarily-invariant ideal E in the algebra L.H / of all bounded linear operators on a Hilbert space H is a (Banach) norm provided this is the case for its restriction to the diagonal subspace of E (that is, on the set of all operators from E which are diagonal with respect to a given orthonormal basis in H ). In my PhD thesis, I answered this question under the additional assumption that the norm is monotone with respect to Hardy-Littlewood [sub]majorization. That was a frustrating restriction and, for many years, I returned again and again to that problem. It was only natural that at some stage in 2006 while we were fi- nalizing our work (begun earlier in Oberwolfach) on singular traces which are not monotone with respect to the Hardy–Littlewood [sub]majorization, I again looked at this infinite-dimensional analogue of von Neumann’s result and explained it to Nigel together with a rather long account of various approaches I had tried in the past. Basically, it took Nigel a couple of weeks and a long flight from Adelaide to the US to come up with an outstanding new idea which we later termed “uniform Hardy–Littlewood [sub]majorization”. This was the key to the solution of that problem. The paper entitled “Symmetric norms and spaces of operators” was published in 2008 by Crelle’s journal and we both believed that this paper will prove itself useful for various questions in analysis of symmetric spaces. One must also bear in mind the extraordinary reticence and modesty (typical for Nigel) and his extreme aversion to boasting. His modesty was as great as his genius. It took some effort from my part to submit that article to a “big journal” and he acquiesced only because we both thought of it as a major contribution to the area. Chapter 4 Symmetric Functionals

4.1 Introduction

In Section 2.7 we introduced the notion of a symmetric functional on a symmetric operator space. We repeat it here for convenience.

Deﬁnition 4.1.1. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace .LetE.M, / be a symmetric operator space. The functional ' 2 E.M, / is called (a) symmetric if '.A/ D '.B/ whenever .A/ D .B/ for 0 A, B 2 E.M, /.

(b) fully symmetric if '.B/ '.A/ whenever .A/ .B/ for 0 A, B 2 E.M, /. The adjective "symmetric" applied to a functional on E.M, / implies automatically that the functional is a continuous linear functional on E.M, /. Another aspect to note from the deﬁnition is that a symmetric functional is not necessarily a positive linear functional whereas a fully symmetric functional is automatically a positive linear functional. Lemma 2.7.4 of Chapter 2 noted that the symmetric functionals describe the continuous traces on a symmetric operator space that can be constructed using a formula on singular values. When M is an atomless (or atomic) factor all continuous traces on E.M, / are symmetric functionals. In particular, when M D L.H / for a separable Hilbert space H , the theory that we present in this chapter accounts for all continuous traces on symmetrically normed ideals of compact operators. Chapter 3 states that the Calkin correspondence is a functor from symmetric function spaces to symmetric operator spaces, which is bijective in the atomless (or atomic) case. It is then natural to ask whether the Calkin correspondence extends to a functor from symmetric functionals on symmetric function spaces to symmetric functionals on symmetric operators spaces (in different terminology, do the symmetric functionals on a function space lift to the corresponding operator space). In particular, if ' 2 E is a symmetric functional on the symmetric function space E,andE.M, / is the symmetric operator space corresponding to E in Theorem 3.5.5, does the assignment

L.'/.A/ :D '..A//,0 A 2 E.M, / deﬁne a symmetric functional on E.M, /, and does the assignment provide a bijective correspondence in the atomless (or atomic) case? The afﬁrmative answer is provided 108 Chapter 4 Symmetric Functionals in Theorem 4.4.1 in Section 4.4. A similar functor from symmetric functionals on symmetric sequence spaces to symmetric functionals on symmetric operator spaces is shown using the construction of Section 3.6. The importance of the functors is that we can answer the following natural existence questions about symmetric functionals on operator spaces by answering the same questions about symmetric functionals on symmetric function (or sequence) spaces.

Question 4.1.2. (a) Which symmetric operator spaces admit a nontrivial symmetric functional?

(b) Which fully symmetric operator spaces admit a nontrivial fully symmetric functional?

(c) Which fully symmetric operator spaces admit a nontrivial symmetric functional that is not a fully symmetric functional?

The crucial component in answering these questions is the Figiel–Kalton theorem concerning the notion of the center of a symmetric function space (and the tool of uniform submajorization introduced in the last chapter). In answering Question 4.1.2 we investigate and describe for the atomless (or atomic) case the class of symmetric operator spaces admitting a symmetric functional. We consider how many symmetrical functionals may be admitted. We show that a symmetric functional decomposes into two Hermitian (or four positive) symmetric functionals and that the set of Hermitian symmetric functionals form a Banach lattice which is either trivial, 1-dimensional or inﬁnite-dimensional. Another result shown in this chapter is that symmetric functionals decompose into normal and singular parts. It is shown that if E.M, / 6 L1.M, / (we are still considering here that M is atomless or atomic), then every symmetric functional ' on E.M, / is singular. To be mundane

' D 0 C 's where 's is a singular symmetric functional and 0 is the trivial normal functional. If E.M, / L1.M, / then

' D ˛ C 's for some constant ˛ and a singular symmetric functional 's .Sothe theory of singular symmetric functionals in the atomless (or atomic) case is the entire theory of symmetric functionals beyond the trivial or canonical normal trace. The following theorem is the main result of this chapter. It yields a complete answer to Question 4.1.2. For brevity, we state the result for the case M D L.H / and in terms of continuous traces (knowing, from Lemma 2.7.4, that all continuous traces Section 4.2 Jordan Decomposition of Symmetric Functionals 109 are symmetric functionals for the factor L.H /). The statement (and the proof) for semiﬁnite atomless or atomic von Neumann algebras can be found in Section 4.9.

Theorem 4.1.3. Let E be a symmetric ideal of L.H /. Consider the following conditions. (a) There exist nontrivial continuous singular traces on E.

(b) There exist nontrivial continuous singular traces on E, which are fully symmetric.

(d) E ¤ L1 and there exists an operator A 2 E such that 1 lim k „A ˚˚ƒ‚ A… kE > 0. (4.1) m!1 m m times

(i) The conditions (a) and (d) are equivalent for every symmetric ideal.

(ii) The conditions (a), (b) and (d) are equivalent for every symmetric ideal that is fully symmetric.

(iii) The conditions (a)–(d) are equivalent for every symmetric ideal that is fully symmetric and equipped with a Fatou norm.

Observe that if E D L1, then the condition (4.1) trivially holds for every 0 ¤ A 2 L1. The ideal of trace class operators L1 admits the continuous trace Tr, so we have the result that a symmetric ideal of compact operators admits a continuous trace if and only if (4.1) holds (see Corollary 4.9.3).

4.2 Jordan Decomposition of Symmetric Functionals

Let M be a von Neumann algebra equipped with a ﬁxed faithful normal semiﬁnite trace and let E.M, / be a symmetric operator space. A functional ' 2 E.M, / is called Hermitian if '.A/ D '.A/ for every A 2 E.M, /.

Deﬁnition 4.2.1. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. The mapping 'C : E.M, /C ! R, deﬁned by setting

'C.A/ :D supf'.B/ :0 B Ag,0 A 2 E.M, /, is called the positive part of the functional '. 110 Chapter 4 Symmetric Functionals

We show below (in Proposition 4.2.8) that the linear extension of 'C is a positive symmetric functional on E.M, /. The main result of this section is the following analog of the Jordan decomposition of a normal form on a von Neumann algebra (see e.g. Theorem 5.17 in [227]). The reader should bear in mind that symmetric functionals are generally singular and, therefore, the classical techniques used on normal functionals are not applicable here.

Theorem 4.2.2. Let M be an atomless or atomic semiﬁnite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a symmetric functional. There exist positive symmetric functionals 'k 2 E.M, / ,1 k 4, such that

' D '1 '2 C i'3 i'4.

The functionals '1 '2 and '3 '4 are Hermitian with positive parts '1 and '3, respectively.

Proof. Let ' be a symmetric functional on E.M, /. The mapping 'N : A ! '.A/ is also a symmetric functional. The equality

1 1 ' D .' CN'/ C i .' N'/ 2 2i is the decomposition of a symmetric functional ' into Hermitian symmetric functionals. We can therefore assume, without loss of generality, that ' is Hermitian. Given that the linear extension of 'C is a positive symmetric functional on E.M, / (see Proposition 4.2.8 below), we deﬁne a functional ' 2 E.M, / by setting ' :D 'C '.Since'C.A/ '.A/ for every positive operator A 2 E.M, /, it follows that ' is also a positive symmetric functional.

The rest of this section proves Proposition 4.2.8.

Lemma 4.2.3. Let M be a semiﬁnite von Neumann algebra with .1/ D1, and let E.M, / be a symmetric operator space such that M E.M, /. Every continuous symmetric functional ' on E.M, / vanishes on M.

Proof. Let q 2 M be a projection such that .q/ D .1 q/ D1.Wehave.q/ D .1 q/ D 1 and, therefore, '.q/ D '.1 q/ D '.1/. It follows that '.1/ D 0. Let p 2 M be a projection. If .p/ D1,then.p/ D 1 and, therefore, '.p/ D '.1/ D 0. If .p/ < 1,then.1 p/ D1 and, therefore, '.1 p/ D '.1/ and '.p/ D 0. Hence, '.p/ D 0 in either case. Recall that every element in M can be uniformly approximated by linear combinations of projections. If M E.M, /, then it follows from the Closed Graph Theorem Section 4.2 Jordan Decomposition of Symmetric Functionals 111

that kkE kk1. Thus, every element in M can be approximated by linear combinations of projections in the norm topology of E.M, /. It now follows from the previous paragraph that the bounded symmetric functional ' vanishes on M.

In the next lemma s , s>0, is the dilation operator on .L1 C L1/.0, 1/.

Lemma 4.2.4. Let M be an atomless or atomic semiﬁnite von Neumann algebra and let E.M, / be a symmetric operator space. Let ' be a symmetric functional on E.M, / and let A1, A2 2 E.M, / be positive operators.

(a) If .A2/ D 1=n.A1/, n 2 N, then '.A2/ D 1=n'.A1/.

(b) If .1/ D1and .A2/ D n.A1/, n 2 N, then '.A2/ D n'.A1/.

N ˚n Proof. Suppose ﬁrst that .A2/ D 1=n.A1/, n 2 . The operator A2 is well ˚n deﬁned and .A1/ D .A2 /. On the other hand,

˚n A2 D A2 ˚ 0 ˚0 C 0 ˚ A2 ˚ 0 ˚0 C0 ˚ 0 ˚A2.

˚n Since ' is a symmetric functional, it follows that '.A2 / D n'.A2/. Again using the fact that ' is symmetric, we obtain that '.A1/ D n'.A2/. This proves the ﬁrst assertion. The second assertion now follows from the fact that 1=nn is the identity operator on l1 (or L1.0, 1/).

The following lemma shows that one can replace the standard order in Deﬁni- tion 4.2.1 with uniform submajorization. This lemma demonstrates the key role played by uniform submajorization for symmetric functionals.

Lemma 4.2.5. Let M be an atomless or atomic semiﬁnite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. We have

'C.A/ D supf'.B/ :0 B C Ag,0 A 2 E.M, /.

Proof. Let 0 B be such that B C A. It follows from Corollary 3.4.3 that B 2 E.M, /.SetD D .B .1, B//C .Since0 D B and 0 B C A, it follows that 0 D C A and .1, D/ D 0. We claim that '.B/ D '.D/. If the algebra M is ﬁnite or if M 6 E.M, /,then .1, B/ D 0. In this case, B D D and the claim follows immediately. Now let M be an inﬁnite von Neumann algebra and let E.M, / M. By Lemma 4.2.3, ' vanishes on M.SinceB D 2 M, the claim follows. 112 Chapter 4 Symmetric Functionals

Fix >0. It follows from Theorem 3.4.2 that there exist positive functions zk, 1 k n, and positive constants k,1 k n, such that .zk/ .A/ and

Xn Xn .1 /.D/ D kzk, k D 1. (4.2) kD1 kD1

Let i be the mapping constructed in Theorem 2.3.11.Set Dk D i.zk / for 1 k n. Since .zk / .A/ and .Dk/ D .zk/,1 k n, it follows that 0 Dk and .Dk/ .A/P for 1 k n. Applying i to both sides of (4.2), we obtain that n .1 /D D kD1 kDk. Therefore,

Xn .1 /'.B/ D .1 /'.D/ D k'.Dk/ supf'.B/ :0 B Ag. kD1

Since >0 is arbitrarily small, the assertion follows.

The following lemma is the ﬁrst step in showing that the linear extension of 'C is a positive linear functional on E.M, /.

Lemma 4.2.6. Let M be an atomless or atomic semiﬁnite von Neumann algebra and E M let A1, A2 2 . , / be positive operators. If .A2/ D 21=2.A1/,then'C.A1/ D 'C.A2/.

Proof. First, we prove that 'C.A1/ 'C.A2/. By the assumption of the lemma, A1 C A2. It follows from Lemma 4.2.5 that

'C.A1/ D supf'.B/ :0 B C A1gsupf'.B/ :0 B C A2gD'C.A2/.

In order to prove the converse inequality, select 0 B A2. Select a positive operator 0 C A1 such that .B/ D 21=2.C /. It follows from Lemma 4.2.4 that '.B/ D '.C/ 'C.A1/. Taking the supremum over B, we obtain 'C.A2/ 'C.A1/. The next lemma is the second step.

Lemma 4.2.7. Let M be an atomless or atomic semiﬁnite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. For all positive operators A1, A2 2 E.M, /, we have

'C.A1 ˚ A2/ D 'C.A1/ C 'C.A2/ provided that A1 ˚ A2 2 E.M, /. Section 4.2 Jordan Decomposition of Symmetric Functionals 113

Proof. If 0 B1 A1 and 0 B2 A2,then0 B1 ˚ B2 A1 ˚ A2. Hence, supf'.B/ :0 B A1 ˚ A2gsupf'.B1 ˚ B2/ :0 B1 A1,0 B2 A2g and, therefore,

'C.A1 ˚ A2/ 'C.A1/ C 'C.A2/. (4.3)

In order to prove the converse inequality, let 0 B A1 ˚A2.Letp be the support projection of A1 ˚ 0andletU D p C i.1 p/.SinceU is unitary, it follows that '.C/ D '.U 1CU/for every operator C 2 E.M, /. Note that

Bp U 1.Bp/U D Bp U 1Bp D .1 C i/.1 p/Bp.

Therefore, '..1p/Bp/ D 0 and, since ' is Hermitian, it follows that '.pB.1p// D 0. Hence, '.B/ D '.pBp/ C '..1 p/B.1 p//.

On the other hand, we have pBp p.A1˚A2/p D A1˚0, .1p/B.1p/ .1p/.A1˚A2/.1p/ D 0˚A2.

Setting B1 :D pBp and B2 :D .1 p/B.1 p/,wehave'.B/ D '.B1 C B2/ and 0 B1 A1 ˚ 0, 0 B2 0 ˚ A2. Therefore,

'C.A/ D supf'.B/ :0 B A1 ˚ A2g

supf'.B1/ C '.B2/ :0 B1 A1 ˚ 0, 0 B2 0 ˚ A2g. Thus,

'C.A1 ˚ A2/ 'C.A1/ C 'C.A2/. (4.4)

The assertion follows from (4.3) and (4.4).

We are now in a position to prove that 'C is additive on the positive part of E.M, / and extends to a symmetric functional.

Proposition 4.2.8. Let M be an atomless or atomic semiﬁnite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. For positive operators A1, A2 2 E.M, /, we have

'C.A1 C A2/ D 'C.A1/ C 'C.A2/ and

'C.A1/ D 'C.A2/ if .A1/ D .A2/.

In particular, 'C extends to a positive symmetric functional on E.M, /. 114 Chapter 4 Symmetric Functionals

Proof. If 0 A1, A2 2 E.M, /,thenalsoA1 ˚ A2 2 E.M, /. Repeating the argument in Lemma 3.4.4, we obtain

C C A1 ˚ A2 A1 C A2 21=2.A1 ˚ A2/. (4.5)

It follows from Lemma 4.2.5 and (4.5) that

'C.A1 ˚ A2/ D supf'.B/ :0 B C A1 ˚ A2g

supf'.B/ :0 B C A1 C A2gD'C.A1 C A2/.

It follows from Lemma 4.2.5 and (4.5) that

'C.A1 C A2/ D supf'.B/ :0 B C A1 C A2g C supf'.B/ :0 B 21=2.A1 ˚ A2/g.

Lemma 4.2.6 now implies that

'C.A1 C A2/ supf'.B/ :0 B C A1 ˚ A2gD'C.A1 ˚ A2/.

We have shown that 'C.A1 C A2/ D 'C.A1 ˚ A2/ and now, applying Lemma 4.2.7, we obtain

'C.A1 C A2/ D 'C.A1 ˚ A2/ D 'C.A1/ C 'C.A2/.

Finally, if .A1/ D .A2/, then the sets f0 B C A1g and f0 B C A2g are identical. Therefore, from Lemma 4.2.5 we obtain the equality 'C.A1/ D 'C.A2/.

4.3 Lattice Structure on the Set of Symmetric Functionals

Knowing that a Hermitian symmetric functional decomposes into positive symmetric functionals allows us to establish an interesting result (actually, from our viewpoint we view it as a remarkable result) that the set of all Hermitian symmetric functionals on a symmetric operator space E.M, /is a Banach lattice with respect to the operations _, ^ deﬁned below. Later, in Theorem 4.10.1, we show that this Banach lattice is either trivial, 1-dimensional or inﬁnite-dimensional.

Deﬁnition 4.3.1. Let E.M, / be a symmetric operator space and let '1, '2 2 E.M, / be Hermitian symmetric functionals. The mappings '1 _'2 : E.M, /C ! R and '1 ^ '2 : E.M, /C ! R are deﬁned by setting

.'1 _ '2/.A/ :D supf'1.B/ C '2.C / :0 B, C , A D B C C g,0 A 2 E.M, /.

.'1 ^ '2/.A/ :D inff'1.B/ C '2.C / :0 B, C , A D B C C g,0 A 2 E.M, /. Section 4.3 Lattice Structure on the Set of Symmetric Functionals 115

Observe that ' _0 is the functional 'C from Deﬁnition 4.2.1. The following lemma extends Proposition 4.2.8.

Lemma 4.3.2. Let E.M, /be a symmetric operator space and let '1, '2 2 E.M, / be Hermitian symmetric functionals on E.M, /. The mappings '1 _ '2 and '1 ^ '2 deﬁned in Deﬁnition 4.3.1 extend to symmetric functionals on E.M, /.

Proof. For every positive A,wehave

.'1 _ '2/.A/ D supf'1.B/ C '2.C / :0 B, C , A D B C C g

D supf'1.A/ C .'2 '1/.C / :0 B, C , A D B C C g

D '1.A/ C supf.'2 '1/.C / :0 C Ag

D '1.A/ C .'2 '1/C.A/.

It follows from Proposition 4.2.8 that '1 _ '2 extends to a symmetric functional on E.M, /C. By Deﬁnition 4.3.1 we have

'1 ^ '2 D..'1 / _ .'2//.

Therefore, '1 ^ '2 is a symmetric functional.

Corollary 4.3.3. Let E.M, / be a symmetric operator space. For every Hermitian symmetric functional ' 2 E.M, /, the mapping j'j :D ' _ .'/ is a positive symmetric functional on E.M, /.

Proof. By Lemma 4.3.2, j'j is a symmetric functional on E.M, /. For every A 0, let B D A and C D 0if'.A/ 0andletB D 0andC D A otherwise. We have

j'j.A/ '.B/ '.C/ Dj'.A/j0.

Hence, j'j is a positive functional.

We require the norm E on E.M, / deﬁned by setting

'E :D sup j'.A/j, ' 2 E.M, / . (4.6) ADA ,kAkE1

We have 'E k'kE 2 'E , ' 2 E.M, / , and, therefore, the norms E and kkE are equivalent. 116 Chapter 4 Symmetric Functionals

Lemma 4.3.4. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. If E is the norm deﬁned in (4.6),then

'E D j'j E .

Proof.Since' is Hermitian, it follows that

'E D sup j'.A/jD sup '.A/. ADA ,kAkE1 ADA ,kAkE1

Therefore,

'E D sup sup '.A/ . D0,kDkE 1 ADA ,jAjDD

On the other hand, we have

sup '.A/ D supf'.AC A/ : AC, A 0, AC C A D DgDj'j.D/. ADA,jAjDD

It follows that

'E D sup j'j.D/ D j'j E . D0,kDkE 1

Theorem 4.3.5. Let E.M, /be a symmetric operator space. The set of all Hermitian symmetric functionals in E.M, / is a Banach lattice with respect to the operations _ and ^ and the norm E .

Proof. Clearly, the set of all Hermitian symmetric functionals on E.M, /is a closed subspace of E.M, / and is, therefore, a Banach space. First, we show that the operations _ and ^ are compatible with the natural order on the Hermitian part of E.M, /. It follows immediately from the deﬁnition of '1_'2 that '1_'2 '1 and '1_'2 '2. Conversely, let ' '1 and ' '2. For every positive A 2 E.M, / and every decomposition A D B C C with positive B, C ,wehave

'.A/ D '.B/ C '.C/ '1.B/ C '2.C /.

Taking the supremum over all such decompositions, we obtain ' '1 _ '2. Recall that j'jD' _ .'/.If'1, '2 2 E.M, / are Hermitian symmetric functionals such that j'2jj'1j, then it follows from Lemma 4.3.4 that

'2E D j'2jE j'1jE D '1 E .

Note that the Hermitian part of the Banach dual E.M, / is not a lattice in general.

Example 4.3.6. There exists a linear Hermitian functional ' 2 M2.C/ such that 'C is nonlinear. Section 4.4 Lifting of Symmetric Functionals 117

Proof. Let 10 11 12 A D , B D , C D . 00 11 24

The matrices A and B=2 are projections. Hence, 0 A1 A implies that A1 D A, 2 Œ0, 1. Therefore, 'C.A/ D maxf'.A/,0g. Similarly, 'C.B/ D maxf'.B/,0g. Clearly, 10.A C B/ C and, therefore, 'C.A C B/ 1=10'.C/. Setting a11 a12 ' : ! a22 a11, a21 a22 we obtain that 'C.A/ D 'C.B/ D 0and'C.A C B/ 3=10.

4.4 Lifting of Symmetric Functionals

In this section, we explain how a symmetric functional on a symmetric function (or sequence) space lifts to a symmetric functional on the corresponding symmetric operator space. In the atomless (or atomic) case we show that the lift is a bijection. The main result is Theorem 4.4.1 below. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace .IfE is a symmetric function space, the corresponding symmetric operator space E.M, / given in Theorem 3.1.2 is

E.M, / DfA 2 S.M, / : .A/ 2 Eg , kAkE Dk.A/kE .

If E is a symmetric sequence space, the corresponding symmetric operator space E.M, / given in Corollary 3.6.7 is

E.M, / DfA 2 S.M, / : E..A/jA/ 2 Eg , kAkE DkE..A/jA/kE .

Here E is the expectation operator defined in Section 3.6 and A :Df.n 1, ngn2N. We reiterate our convention that an atomic semifinite von Neumann algebra satisfies the condition that .p/ D 1 for any non-zero minimal projection p 2 Proj.M/,and that, in this case, E..A/jA/ D .A/.

Theorem 4.4.1. Let E be a symmetric function (respectively, sequence) space and let E.M, / be the corresponding symmetric operator space as above. (a) If ' is a symmetric functional on E, then there exists a symmetric functional L.'/ on E.M, / such that L.'/.A/ D '..A// 118 Chapter 4 Symmetric Functionals

(respectively, L.'/.A/ D '.E..A/jA//

for all positive A 2 E.M, /. The functional L.'/ is positive if the functional ' is positive.

(b) Let M be an atomless (or atomic) semiﬁnite von Neumann algebra. If ' is a symmetric functional on E.M, /, then there exists a symmetric functional L1.'/ on E such that '.A/ D L1.'/.x/

for all positive x 2 E and A 2 E.M, / such that .A/ D .x/. The functional L.'/ is positive if and only if the functional ' is positive.

The theorem is proved below. Observe that the next corollary follows immediately from Theorem 4.4.1.

Corollary 4.4.2. Let E and E.M, / be as in Theorem 4.4.1. (a) The functional L.'/ is a fully symmetric functional on E.M, / if ' is a fully symmetric functional on E.

(b) If M is atomless (or atomic), then L.'/ is a fully symmetric functional on E.M, / if and only if ' is a fully symmetric functional on E.

We now prove Theorem 4.4.1. We start with an intermediate result from Section 4.2.

Lemma 4.4.3. Let E be a symmetric function (or sequence) space and let ' be a positive symmetric functional on E.

(a) If x, y 2 EC are such that y C x, then '.y/ '.x/.

(b) If x 2 E then '..x// D '.21=2.x//.

Proof. The ﬁrst part is immediate from Lemma 4.2.5 since '.y/ 'C.x/ D '.x/. The second part is Lemma 4.2.4.

We can lift symmetric functionals between a symmetric sequence space E and the symmetric function space

F :Dfx 2 .L1 C L1/.0, 1/ : E..x/jA/ 2 Eg , kxkF :DkE..x/jA/kE constructed in Section 3.6. We identify the sequence space E with the subspace in F spanned by the characteristic functions of the partition A Df.n 1, ngn2N. Section 4.4 Lifting of Symmetric Functionals 119

Theorem 4.4.4. Let E be a symmetric sequence space and let F be the symmetric function space as above. (a) Every symmetric functional on E extends to a symmetric functional on F .

(b) The restriction of every symmetric functional on F to E is a symmetric functional.

Proof. Let ' be a positive symmetric functional on E.Set

'.x/ D '.E..x/jA//,0 x 2 F .

If 0 x, y 2 F , then it follows from Lemma 3.6.4 that

E A C E A C E A ..x C y/j / ..x/ C .y/j / 21=2 ..x C y/j /.

It follows from Lemma 4.4.3 that

'.E..x C y/jA// D '.E..x/ C .y/jA//.

Thus, ' is additive on the positive part of F . Hence, ' admits an extension as a continuous linear functional on F . Evidently, ' is symmetric. In general, (a) follows from the above argument and the decomposition of a general symmetric functional into a linear combination of positive symmetric functionals as shown in Theorem 4.2.2. The proof of (b) is trivial.

Proof of Theorem 4.4.1.

(a) Let E be a symmetric function space. Let A, B 2 E.M, /C. It follows from Lemma 3.4.4 that

C C .A C B/ .A/ C .B/ 21=2.A C B/.

If the functional ' is positive, then it follows from Lemma 4.4.3 that

'..A C B// '..A/ C .B// '.21=2.A C B// D '..A C B//.

Thus, L.'/ is additive on E.M, /C and, therefore, it admits an extension as a linear functional on E.M, /. Evidently, this functional is symmetric. The assertion is proved for the case of a positive functional '. This is sufﬁcient as, by Theorem 4.2.2, a general symmetric functional on E decomposes into a linear combination of positive symmetric functionals. If E is a symmetric sequence space then, due to Theorem 4.4.4, the argument is reduced to the consideration of the function space F . (b) The proof of (b) is identical to (a) since, due to Theorem 3.1.1, the Calkin correspondence is bijective in the atomless or atomic case. 120 Chapter 4 Symmetric Functionals

4.5 Figiel–Kalton Theorem

Theorem 4.4.1 establishes that the Calkin correspondence provides a lift from symmetric functionals on symmetric function (or sequence) spaces to symmetric functionals on symmetric operators spaces, which is bijective in the atomless (or atomic) case. This important result is central to the theory of continuous traces, since we can use the classical Banach space theory of symmetric function spaces to answer questions concerning the existence of continuous traces. The Figiel–Kalton theorem is a fundamental result that can be used to show the existence of symmetric functionals on symmetric function spaces; we use it in subsequent sections. Let E be a symmetric function space either on the interval .0, 1/ or on the semi-axis. Deﬁne the sets

DE :D Lin.fx 2 E : x D .x/g/ Df.a/ .b/, a, b 2 Eg,

ZE :D Lin.fx1 x2 :0 x1, x2 2 E, .x1/ D .x2/g/.

The set ZE is the common kernel of all linear functionals f : E ! C such that f.x1/ D f.x2/ if .x1/ D .x2/. It is not quite the common kernel of all symmetric functionals since we have not demanded that f is continuous. The Figiel–Kalton theorem gives a description of ZE in terms of the Cesàro operator C : .L1 C L1/.0, 1/ ! S.0, 1/, Z 1 t .Cx/.t/ D x.s/ds, x 2 .L1 C L1/.0, 1/. t 0

Theorem 4.5.1 (Figiel–Kalton). Let E be a symmetric function space on the semi- axis and let x 2 DE . We have x 2 ZE if andR only if Cx 2 E. The same assertion is 1 also valid for the interval .0, 1/ provided that 0 x.s/ds D 0.

The theorem is proved below. Lemma 4.5.2 and Lemma 4.5.3 provide the proof of the “only if” part of Theorem 4.5.1.

Lemma 4.5.2. Let E be a symmetric function space. We have

x 2 ZE H) C..xC/ .x// 2 E. P n Proof.Letx D kD1.xk yk/ with xk, yk 2 EC and .xk/ D .yk/,1 k n. Set Xn Xn z D xC C yk D x C xk. kD1 kD1 Section 4.5 Figiel–Kalton Theorem 121

It follows from the deﬁnition of C and Lemma 3.3.3 that Xn Xn C.z/ C.xC/ C C.yk/ D C..xC/ .x// C C.x/ C C.xk/. kD1 kD1

Using Lemma 3.3.5, we obtain that Z Z Z t Xn .nC1/t t ..s, x/ C .s, xk //ds .s, z/ds .s, z/ds C nt.t, z/. 0 0 0 kD1

Therefore,

C.z/ C.z/ C C..xC/ .x// C n.z/.

It follows that C..x/ .xC// n.z/. Similarly, C..xC/ .x// n.z/ and we conclude that

jC..xC/ .x//jn.z/ 2 E.

Lemma 4.5.3. Let E be a symmetric function space. We have

x 2 DE H) C..xC/ .x// 2 Cx C E.

Proof. It follows from the deﬁnition of DE that x D .a/ .b/ for some a, b 2 E. Observe that

.a/ .a/supp.xC/ xsupp.xC/ D xC.

Set u :D .a/ xC 0. Clearly, .a/ D u C xC and .b/ D u C x. It follows from the deﬁnition of C and Lemma 3.3.3 that

C.a/ C.u/ C C.xC/ D C..xC/ .x// C C.u/ C C.x/.

Using Lemma 3.3.5, we obtain that

C.x/ C C.u/ C.b/ C .b/.

It follows that

Cx C..xC/ .x// C .b/.

Similarly,

Cx C..xC/ .x// .a/, and the assertion follows. 122 Chapter 4 Symmetric Functionals

We now proceed with Lemma 4.5.5, which provides the proof of the “if” part of Theorem 4.5.1. We need the following deep theorem due to Kwapien [141]. We omit the proof of Theorem 4.5.4 as it falls beyond the scope of this book. R 1 Theorem 4.5.4. For every x 2 L1.0, 1/ such that 0 x.s/ds D 0, there exist positive functions y1, y2 2 L1.0, 1/ such that .y1/ D .y2/, x D y1 y2 and ky1k1, ky2k1 6kxk1.

Lemma 4.5.5. Let E be a symmetric function space on .0, 1/ and let x 2 DE . If Cx 2 E, then x 2 ZE . The same assertion holds forR such an element x of a symmetric 1 function space on the interval .0, 1/ provided that 0 x.s/ds D 0. n nC1 Proof. Deﬁne a partition A :Df.2 ,2 gn2Z and set x1 :D E.xjA/.Ifx D .a/ .b/ with a, b 2 E,thenx1 D E..a/jA/ E..b/jA/. Clearly,

E..a/jA/ 2.a/ 2 E, E..b/jA/ 2.b/ 2 E are decreasing functions. It follows that x1 2 DE . It is easy to see that

jCx1 Cxj22..a/ C .b//.

Since .a/, .b/, Cx 2 E, it follows that Cx1 2 E. Deﬁne a function z 2 E by setting nC1 n nC1 z.t/ :D .Cx1/.2 /, t 2 .2 ,2 /.

Clearly, x1 D 2z 2z 2 ZE . n nC1 Consider the function x x1 on the interval .2 ,2 /. By Theorem 4.5.4, there n nC1 exist positive functions y1n, y2n supported on .2 ,2 / such that

.x x1/.2n,2nC1/ D y1n y2n and

.y1n/ D .y2n/, ky1nk1, ky2nk1 6k.x x1/.2n,2nC1/k1.

Set X X y1 :D y1n, y2 :D y2n. n2Z n2Z

It follows from the above inequality that y1, y2 2 EC.Sincex x1 D y1 y2 and .y1/ D .y2/, it follows that x x1 2 ZE . Since we already established that x1 2 ZE , it follows that x 2 ZE .

Proof of Theorem 4.5.1. If x 2 ZE \ DE , then it follows from Lemma 4.5.2 that C..xC/ .x// 2 E. By Lemma 4.5.3, we have that Cx 2 E. Section 4.6 Existence of Symmetric Functionals 123

Conversely, if x 2RDE (if E is symmetric function space on the interval .0, 1/,then 1 we require also that 0 x.s/ds D 0) and Cx 2 E, then it follows from Lemma 4.5.5 that x 2 ZE .

4.6 Existence of Symmetric Functionals

In this section, we present results concerning the existence of (singular) symmetric functionals on symmetric function spaces. In Section 4.9 we spell out the consequences of the existence results shown in this section for the existence of symmetric functionals on symmetric operator spaces.

Theorem 4.6.1. Let E be a symmetric function space and let 0 x 2 E. (a) If E D E.0, 1/, then there exists a positive symmetric functional ' 2 E such that 1 '.x/ D lim km.x/kE . m!1 m

(b) If E D E.0, 1/ L1.0, 1/, then there exists a positive singular symmetric functional ' 2 E such that

1 '.x/ D lim k.m.x//.0,1/kE . m!1 m

(c) If E D E.0, 1/, then there exists a positive singular symmetric functional ' 2 E such that 1 '.x/ D lim km.x/kE . m!1 m

Theorem 4.6.1 is proved below. The next result is that every symmetric functional on a symmetric function space decomposes into normal and singular symmetric functionals.

Theorem 4.6.2. Let E be a symmetric function space and let ' 2 E be a symmetric functional.

(a) If E D E.0, 1/ 6 L1.0, 1/, then ' is singular.

(b) If E D E.0, 1/ L1.0, 1/, or E D E.0, 1/, then there exists a singular symmetric functional 's 2 E such that ' D ˛m C 's for a scalar ˛.Herem denotes the Lebesgue integral. To prove the theorems we need the following variant of the Hahn–Banach theorem for partially ordered spaces, and the subsequent lemmas. 124 Chapter 4 Symmetric Functionals

Lemma 4.6.3. Let E be a partially ordered linear space and let p : E ! R be a convex and monotone functional (that is, p.x/ p.y/ provided that x y and x, y 2 E). For every x0 2 E, there exists a positive linear functional ' : E ! R such that ' p and '.x0/ D p.x0/.

Proof. The existence of a linear functional ' p such that '.x0/ p.x0/ follows from the Hahn–Banach theorem. We only have to prove that ' 0. If z 0, then, by assumption, p.x0/ p.x0 z/. Therefore,

'.z/ D '.x0/ '.x0 z/ D p.x0/ '.x0 z/ p.x0/ p.x0 z/ 0.

Recall that the sets DE and ZE were introduced at the beginning of Section 4.5.

Lemma 4.6.4. Let E be a symmetric function space on the semi-axis. If the mapping : E ! E is deﬁned by setting

.x/ :D .xC/ .x/, then

(a) : E ! DE and : ZE ! DE \ ZE . For every x 2 E, we have .x/ 2 x C ZE . (b) for every x, y 2 E, we have

.x C y/ 2 .x/ C .y/ C DE \ ZE .

Proof. Wederive(b)from(a). (a) The ﬁrst assertion is trivial. For every x 2 E, it follows from the deﬁnition of ZE that .xC/ xC 2 ZE and .x/ x 2 ZE . Hence,

.x/ x D ..xC/ xC/ ..x/ x/ 2 ZE .

This proves the third assertion. The second assertion follows by observing that .x/, x 2 E belongs to DE and that .x/ 2 ZE CZE D ZE for every x 2 ZE . (b) It follows from (a) that

.x C y/ 2 x C y C ZE , .x/ 2 x C ZE , .y/ 2 y C ZE

and, therefore,

.x C y/ 2 .x/ C .y/ C ZE .

The result is shown since : E ! DE . Section 4.6 Existence of Symmetric Functionals 125

Lemma 4.6.5. Let E be a symmetric function space on the semi-axis. Let p : DE ! R be a convex and monotone functional. If p D 0 on ZE \ DE , then

(a) p D p ı on DE , where is the mapping in Lemma 4.6.4.

(b) p extends to a convex monotone functional p : E ! R by setting p D p ı .

Proof. It is stated in Lemma 4.6.4 (a) that .x/ x 2 ZE for every x 2 E.Fromthe same lemma we also have that .x/ 2 DE . Thus, .x/ x 2 DE for every x 2 DE and we conclude that .x/ D x C z for some z 2 DE \ ZE . It follows from the assumptions that

p..x// p.x/ C p.z/ D p.x/, p.x/ p..x// C p.z/ D p..x//.

This proves (a). It follows from Lemma 4.6.4 that .xC y/ D .x/C .y/C z with z 2 DE \ZE for every x, y 2 E. It follows from the assumptions that

p..x C y// p..x// C p..y// C p.z/ D p..x// C p..y//.

This proves the convexity of the extension p ı . Since both the functional p : DE ! R and the mapping : E ! E are monotone, it follows that the extension p ı is also monotone. This proves (b). If x 2 ZE ,then.x/ 2 DE \ ZE . Thus, p.x/ D p..x// D 0, which proves (c).

Lemma 4.6.6. If 0 x 2 .L1 C L1/.0, 1/, then Z Z Z b=m 1 b 1 b x.s/ds 1 m Cx .s/ds x.s/ds a log.m/ a m a=m provided that ma b. In particular, 1 1 m1 x C 1 Cx C x m log.m/ m m provided that x D .x/.

Proof. Clearly, Z 1 1 1 t 1 m Cx .t/ D x.s/ds, t>0. log.m/ m t log.m/ t=m 126 Chapter 4 Symmetric Functionals

Therefore, Z Z Z b 1 b t dt 1 Cx .s/ds D x.s/ds m m t a Za t=mZ b minfms,bg dt D x.s/ds f g t Za=m max a,s b minfms, bg D x.s/log ds. a=m maxfa, sg The integrand does not exceed x.s/log.m/ and the second inequality follows immediately. The integrand is positive and is equal to x.s/log.m/ for s 2 .a, b=m/.The ﬁrst inequality follows.

Lemma 4.6.7. Let E be a symmetric function space on the semi-axis. The functional p : DE ! R deﬁned by setting 1 1 p.x/ :D lim sup 1 m Cx , x 2 DE , m!1 log.m/ m C E satisﬁes the assumptions of Lemma 4.6.5. Also, for every x 2 DE , we have p.x/ kxkE .

Proof. If x 2 DE ,thenx D .a/ .b/ for some a, b 2 E.Setz D .a/ C .b/ 2 E. It follows from Lemma 4.6.6 and Corollary 3.4.3 that 1 1 1 m Cx 1 m Cz log.m/ kzkE . m C E m E

Thus, p.x/ is ﬁnite for all x 2 DE . Observe that the mappings, m 1, 1 x ! 1 m Cx , x 2 E, m C are convex and monotone. So are the mappings 1 x ! 1 m Cx , x 2 E. m C E

Therefore, p : DE ! R is a convex and monotone functional. If x 2 ZE \ DE , then, by Theorem 4.5.1, we have that Cx 2 E. Therefore, 1 1 1 m Cx kCxkE C kmCxkE 2kCxkE. m C E m Letting m !1, we obtain p.x/ D 0. Section 4.6 Existence of Symmetric Functionals 127

The following lemma shows that every symmetric functional on E.0, 1/ 6 L1.0, 1/ is necessarily a singular symmetric functional.

Lemma 4.6.8. If the symmetric function space E on the semi-axis is such that E 6 L1.0, 1/, then every symmetric functional ' on E is singular.

Proof. Let x 2 E be bounded and ﬁnitely supported. We have .x/ kxk1.0,n/ for some n 2 N. Therefore, ˇ ˇ ˇ ˇ ˇ 1 ˇ 1 j'.x/jDˇ' m.x/ ˇ k'kE km.x/kE m m 1 k'k kxk1 k k . E m .0,nm/ E

Letting m !1and taking into account that E 6 L1.0, 1/, we obtain that the right- hand side tends to 0. Therefore, '.x/ D 0 for every bounded and ﬁnitely supported x. By continuity, '.x/ D 0 for every x 2 .L1 \ L1/.0, 1/. Hence, the functional ' is singular.

The case E L1 requires a more detailed treatment.

Lemma 4.6.9. Let E be a symmetric (respectively, fully symmetric) function space either on the interval .0, 1/ or on the semi-axis. Let f'i gi2I 2 E be a net and let ' 2 E be such that 'i ! ' in the weak topology.

(a) If every 'i is symmetric, then ' is symmetric.

(b) If every 'i is fully symmetric, then ' is fully symmetric.

Proof. Let each 'i be a symmetric functional on E.If0 x1, x2 2 E are such that .x1/ D .x2/,then

'.x1/ D lim 'i .x1/ D lim 'i .x2/ D '.x2/. i2I i2I

Hence, ' is a symmetric functional on E. Let each 'i be a fully symmetric functional on E.If0 x1, x2 2 E are such that .x2/ .x1/,then

'.x2/ D lim 'i .x2/ lim 'i .x1/ D '.x1/. i2I i2I

Hence, ' is a fully symmetric functional on E.

Lemma 4.6.10. Let E be a symmetric (respectively, fully symmetric) function space and let ' be a positive symmetric (respectively, fully symmetric) functional on E. The 128 Chapter 4 Symmetric Functionals formula

'sing.x/ :D lim '..x/.0,1=n//,0 x 2 E n!1 deﬁnes a positive singular symmetric (respectively, fully symmetric) linear functional on E.

Proof. If x, y 2 E are positive functions, then it follows from Lemma 3.4.4 that

C C .x C y/.0,1=n/ ..x/ C .y//.0,1=n/ 21=2..x C y/.0,2=n//.

Taking the limit as n !1, we derive from Lemma 4.4.3 that

'sing..x C y// D 'sing..x/ C .y//.

Since ' is symmetric, it follows that

'sing.x C y/ D 'sing..x C y// D 'sing..x/ C .y// D 'sing.x/ C 'sing.y/.

Hence, 'sing is an additive functional on EC. Therefore, it extends to a linear functional on E. Clearly, 'sing is symmetric. If ' is fully symmetric, then the fact that 'sing is fully symmetric is trivial.

Lemma 4.6.11. Let E be a symmetric function space on the semi-axis and let ' be a positive symmetric functional on E. If E L1.0, 1/ and if 'sing is the functional deﬁned in Lemma 4.6.10,then' 'sing is a scalar multiple of the Lebesgue integral.

Proof. Fix a positive function z 2 .L1 \ L1/.0, 1/.For>0, select a positive step function u 2 .L1 \ L1/.0, 1/ such that kz ukL1\L1 .Sinceu is a step function, it follows that '.u/ Dkuk1'..0,1//.Since' is a continuous function, it follows that

j'.z/ kzk1'..0,1//jj'.z u/jCku zk1'..0,1// k'kE ku zkE

Cku zk1 k'kE 2k'kE ku zkL1\L1 2k'kE .

Since is arbitrarily small, it follows that '.z/ Dkzk1'..0,1//. It now follows from the deﬁnition of 'sing that, for every positive x 2 E, Z 1 .' 'sing/.x/ D lim '..x/.1=n,1// D '..0,1// .s, x/ds. n!1 0

The assertion follows from the linearity of ' 'sing. Now, we prove Theorems 4.6.1 and 4.6.2. Section 4.6 Existence of Symmetric Functionals 129

Proof of Theorem 4.6.1 (a). Without loss of generality, x D .x/.Letp be the convex monotone functional constructed in Lemma 4.6.7. It follows from Lemma 4.6.3 that there exists a positive linear functional ' on E such that ' p and '.x/ D p.x/. Since p.z/ D 0 for every z 2 ZE , it follows that '.z/ D 0 for every z 2 ZE . Therefore, ' is a symmetric functional. For every z D .z/ 2 E, it follows from Lemma 4.6.6 and Corollary 3.4.3 that p.z/ kzkE . Hence, '.z/ p.z/ kzkE for every z D .z/ 2 E. It follows that k'kE 1. Therefore, 1 1 '.x/ D ' x k xk . m m m m E Taking the limit m !1, we obtain 1 '.x/ lim km.x/kE . m!1 m On the other hand, it follows from Lemma 4.6.6 that 1 1 m1 x C 1 Cx. m log.m/ m m Therefore, 1 1 1 p.x/ D lim sup 1 m Cx lim km.x/kE . m!1 m!1 log.m/ m E m The assertion follows immediately.

Proof of Theorem 4.6.1 (b). Fix j 1 and apply Theorem 4.6.1 (a) to the function .x/.0,1=j / . It follows that there exists a symmetric linear functional 'j such that k'j kE 1and 1 1 'j ..x/.0,1=j / / D lim km..x/.0,1=j / /kE lim km..x//.0,1/kE . m!1 m m!1 m Since the unit ball in E is weak compact (the Banach–Alaoglu theorem), there exists I N a convergent subnet i D 'F.i/, i 2 , of the sequence 'j , j 2 .Let i ! ' in the weak topology. It follows from Lemma 4.6.9 that ' is a symmetric functional. By the deﬁnition of a subnet (see [188, Section IV.2]), for every ﬁxed j 2 N,there exists ij 2 I such that F.i/ > j for every i>ij . Thus, for every ij

The subnet i , ij

Now, taking the limit as j !1and using Lemma 4.6.10, we obtain the inequality 1 'sing.x/ lim km..x//.0,1/kE , m!1 m where 'sing is the singular symmetric functional deﬁned in Lemma 4.6.10. The op- posite inequality can be obtained by the same argument as in the proof of Theo- rem 4.6.1 (a).

Proof of Theorem 4.6.1 (c). Let F be the symmetric function space (on the semi-axis) consisting of all measurable functions x on .0, 1/ that are ﬁnite for the norm given by the formula

kxkF :Dk.x/.0,1/kE Ckxk1.

Clearly, F L1.0, 1/. Applying Theorem 4.6.1 (b), we obtain a singular symmetric functional ' on F such that 1 1 '.x/ D lim km..x//.0,1/kF D lim km..x//kE . m!1 m m!1 m Proof of Theorem 4.6.2 (a). Lemma 4.6.8 provides the result for positive symmetric functionals. We then use the Jordan decomposition in Theorem 4.2.2. Proof of Theorem 4.6.2 (b). Lemma 4.6.11 provides the result for positive symmetric functionals. We then use the Jordan decomposition in Theorem 4.2.2.

4.7 Existence of Fully Symmetric Functionals

In this section we consider the existence of fully symmetric functionals on fully symmetric function spaces. In Theorem 4.7.1 the reader will notice identical existence criteria to Theorem 4.6.1 in the last section. Again, the consequences of the results of this section for the existence of fully symmetric functionals on fully symmetric operator spaces are discussed in Section 4.9.

Theorem 4.7.1. Let E be a fully symmetric function space and let 0 x 2 E. (a) If E D E.0, 1/, then there exists a fully symmetric functional ' 2 E such that 1 '.x/ D lim km.x/kE . m!1 m

(b) If E D E.0, 1/ L1.0, 1/, then there exists a singular fully symmetric functional ' 2 E such that 1 '.x/ D lim k.m.x//.0,1/kE . m!1 m Section 4.7 Existence of Fully Symmetric Functionals 131

The following lemmas, in combination with some of those in the last section, prove the theorem.

Lemma 4.7.2. If x D .x/ and y D .y/ are such that y x, then 1 1 1 Cy 1 Cx. m m m m

Proof. Arguing as in Lemma 4.6.6, we have that Z Z t 1 t minfms, tg 1 m Cz .s/ds D z.s/log ds. 0 m 0 s For a ﬁxed t>0, the mapping minfms, tg s ! log , s>0 s is decreasing. It now follows from [139, Equality 2.36] that Z Z t 1 t 1 1 m Cy .s/ds 1 m Cx .s/ds 0 m 0 m for every t>0. Since the functions 1 1 1 Cy, 1 Cx m m m m are both decreasing, the assertion follows.

Lemma 4.7.3. Let E be a fully symmetric function space on the semi-axis and let x D .x/ 2 E. If p is the convex functional deﬁned in Lemma 4.6.7 and if z 2 DE is such that Cx Cz, then p.x/ p.z/.

Proof. By assumption, z D .a/ .b/ with a, b 2 E.SinceCx Cz, it follows that C.x C .b// C.a/ or, equivalently, x C .b/ .a/. It follows from Lemma 4.7.2 that, for every t>0, Z Z t 1 t 1 1 m C.x C .b// .s/ds 1 m C.a/ .s/ds 0 m 0 m 132 Chapter 4 Symmetric Functionals and, therefore, Z Z t 1 t 1 1 Cx .s/ds 1 Cz .s/ds m m m m 0 Z0 t 1 1 Cz .s/ds m m Z0 C t 1 s, 1 m Cz ds. 0 m C

The assertion now follows from the deﬁnition of the functional p (given in Lem- ma 4.6.7).

Lemma 4.7.4. Let E be a fully symmetric function space on the semi-axis. Let p be the convex functional deﬁned in Lemma 4.6.7. The functional

q.x/ :D inffp.z/ : z 2 DE , Cx Czg, x 2 DE satisﬁes the assumptions of Lemma 4.6.5.

Proof. Let x1, x2 2 DE .Fix>0 and select z1, z2 2 DE such that Cxi Czi and p.zi / q.xi / C for i D 1, 2. Thus, C.x1 C x2/ C.z1 C z2/ and, by convexity of the functional p,

q.x1 C x2/ p.z1 C z2/ p.z1/ C p.z2/ q.x1/ C q.x2/ C 2.

Since is arbitrarily small, the convexity of the functional q follows. Let x1, x2 2 DE be such that x1 x2.Fix>0 and select z 2 DE such that Cx2 Cz and p.z/ q.x2/ C . Thus, Cx1 Cx2 Cz and q.x1/ p.z/ q.x2/ C .Since is arbitrarily small, it follows that the functional q is monotone. For x 2 ZE \ DE ,wehave0 q.x/ p.x/ D 0 and, therefore, q.x/ D 0.

Proof of Theorem 4.7.1 (a). Without loss of generality, x D .x/.Letq be the convex monotone functional constructed in Lemma 4.7.4. It follows from Lemma 4.6.3 that there exists a positive linear functional ' on E such that ' q and '.x/ D q.x/. By the construction of the functional q,wehaveq p and, therefore ' p.Since p.z/ D 0 for every z 2 ZE , it follows that '.z/ D 0 for every z 2 ZE . Therefore, ' is a symmetric functional. Let x1, x2 2 E be positive elements such that x1 x2. Therefore, z D .x1/ .x2/ 2 DE and Cz 0. It follows from the workings above that '.z/ q.z/ p.0/ D 0. Hence, ' is a fully symmetric functional. For every z D .z/ 2 E, it follows from Lemma 4.6.6 and Corollary 3.4.3 that p.z/ kzkE . Hence, '.z/ q.z/ p.z/ kzkE for every z D .z/ 2 E.It Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 133

follows that k'kE 1. Therefore, 1 1 '.x/ D ' x k xk . m m m m E Taking the limit m !1, we obtain 1 '.x/ lim km.x/kE . m!1 m On the other hand, it follows from the fact that x D .x/ and Lemma 4.7.3 that q.x/ D p.x/. By Lemma 4.6.6, we have 1 1 m1 x C 1 Cx. m log.m/ m m Therefore, 1 1 1 '.x/ D q.x/ D p.x/ D lim sup 1 m Cx lim km.x/kE . m!1 m!1 log.m/ m E m The assertion follows immediately.

The proofs of Theorem 4.7.1 (b),(c) are very similar to that of Theorem 4.6.1 and are, therefore, omitted. The only difference is that the reference to Theorem 4.6.1 (a) has to be replaced with the reference to Theorem 4.7.1 (a).

4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different

The last section gave criteria for the existence of a fully symmetric functional on a fully symmetric function space. Are there symmetric functionals that are not fully symmetric on a fully symmetric function space? In Theorem 4.8.1 we demonstrate that the sets of symmetric and fully symmetric functionals on a given fully symmetric space E equipped with a Fatou norm are distinct provided that a nontrivial singular symmetric functional exists.

Theorem 4.8.1. Let E be a fully symmetric function space equipped with a Fatou norm and let 0 x 2 E. Let '.y/ '.x/ for every 0 y x and for every positive symmetric functional ' 2 E. 1 (a) If E D E.0, 1/ 6 L1.0, 1/, then m m.x/ ! 0 in the norm topology of E.

1 (b) If E D E.0, 1/ L1.0, 1/, then m .m.x//.0,1/ ! 0 in the norm topology of E.

1 (c) If E D E.0, 1/, then m m.x/ ! 0 in the norm topology of E. 134 Chapter 4 Symmetric Functionals

The theorem says, in combination with Theorem 4.7.1 and Theorem 4.2.2, that if every symmetric functional on E is a fully symmetric functional, then all symmetric functionals are trivial. By contradiction then, if a nontrivial fully symmetric functional exists then there must be a distinct symmetric functional. The following lemma is the key technical step in separating fully symmetric functionals from symmetric ones.

Lemma 4.8.2. Let E be a fully symmetric function space. If x, y 2 E are such that '.y/ '.x/ for every positive symmetric functional ' 2 E, then Z Z b mb .s, y/ds ..s, x/ C um.s// ds, ma < b, ma a where 0 um ! 0 in E.

Proof. Without loss of generality, x D .x/ and y D .y/.Letp be a convex positive functional constructed in Lemma 4.6.7. By Lemma 4.6.3, there exists a positive functional ' 2 E such that ' p and '.y x/ D p.y x/. By construction of p,wehavep.z/ D 0 for every z 2 ZE and, therefore, '.z/ D 0foreveryz 2 ZE . Therefore, ' is a positive symmetric linear functional on E. By assumption, '.y x/ 0. Therefore, p.y x/ D '.y x/ 0. Since p takes only positive values, it follows that p.y x/ D 0. Hence, by the deﬁnition of p,we have 1 1 um :D 1 m C.y x/ ! 0 log.m/ m C in E. It follows from the deﬁnition of the convex functional p (see Lemma 4.6.7) that 1 1 1 1 1 Cy 1 Cx C u . (4.7) log.m/ m m log.m/ m m m

By Lemma 4.6.6, we have Z Z b 1 mb 1 y.s/ds 1 m Cy .s/ds. ma log.m/ ma m It now follows from (4.7) and Lemma 4.6.6 that Z Z b mb 1 1 y.s/ds u C 1 Cx .s/ds m log.m/ m m ma Zma mb .x C um/.s/ds. a Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 135

For each sequence and >0, we deﬁne the sequence by setting ( .n/, .n/ .n/ :D 1, .n/ < .

Z Below, we equip the set .RC [f1g/ with componentwise partial ordering.

Z Proposition 4.8.3. Let the mappings Ãm : .RC [f1g/ ! RC, m 1, be quasi- 0 decreasing (that is, there exists a constant const such that implies Ãm./ 0 const Ãm. /). Assuming that

(a) for every m 1, we have Ãm.n/ ! Ãm./ when n #

RZ Ã Ã R Z (b) there exists 0 2 C such that m./ D m. ^ 0/ for all 2 . C [f1g/ , m 1

Z m (c) for every 2 .RC [f1g/ , we have Ãm. / ! 0 as m !1

Z (d) for every 2 .RC [f1g/ , the sequence Ãm./, m 1, decreases we have Ãm.m/ ! 0 as m !1.

Proof. By the assumptions, we have that m (b) m (a) (b) Ãm D Ãm ^ 0 ! Ãm.m ^ 0/ D Ãm.m/, r !1. .r,r/ .r,r/

For every m 1, select rm sufﬁciently large so that m 1 Ãm Ãm.m/. (4.8) .rm,rm/ 2

RZ Without loss of generality, rm "1as m !1. Now deﬁne the element 2 C by setting ^ m :D . m1 . rm,rm/

Clearly, .n/ > m for jnjrm and, therefore .n/ !1as jnj!1. Fix >0andselect such that Ã. /<(this can be done due to (c)). The set fn : .n/ < g is ﬁnite. For an arbitrary m such that

m>maxf,max0.n/g, .n/< 136 Chapter 4 Symmetric Functionals

Ã we have ^ 0 m=.rm,rm/.Since is quasi-decreasing, it follows from (4.8) that (d) m Ãm.m/ 2Ã const Ã. / const . .rm,rm/

Let x D .x/ 2 .L1 C L1/.0, 1/ (or x D .x/ 2 L1.0, 1/)andlet Z t X.t/ :D x.s/ds. 0 For every n 2 Z,leta.n/ be such that

X.a.n// D .3=2/n.

Given a sequence 2 NZ,let

2 B Df.n/a.3n/ :wheren 2 Z is such that .n/ a.3n/ < a.3n C 1/g.

Below we use the expectation operator E.jA/ introduced in Section 3.6. We identify a discrete subset of the semi-axis A with the sets of ﬁnite measure in the partition A [f0g of the semi-axis. The following lemma contains the technically hardest part of the proof of Theo- rem 4.8.1.

Lemma 4.8.4. Let E be a fully symmetric function space. If x D .x/ 2 E is such that '.y/ '.x/ for every positive symmetric functional ' on E and every 0 y x, then, for every sequence , 1 E.xjB 100m / ! 0, m !1 m m in the norm topology of E. Proof. If is a bounded sequence, then the left-hand side is 0 for sufﬁciently large m and the assertion is trivial. Now let be an unbounded sequence. Set y D E.xjB / and let um be the functions constructed in Lemma 4.8.2. Let n 2 Z be such that .n/2a.3n/ < a.3n C 1/ and .n/ 100m. It follows that Z Z Z m.n/a.3n/ m.n/a.3n/ a.3nC1/ um.s/ds .x C um/.s/ds x.s/ds. (4.9) 0 a.3n/ a.3n/

Let n0 be the largest integer number (or 1, if none) such that n0

X..n/a.3n// X..n0/a.3n0// X.a.3n// X.a.3n 2// E.xjB / D (4.10) .n/a.3n/ .n0/a.3n0/ .n/a.3n/ Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 137 on the interval .ma.3n/, .n/a.3n//. It follows from Lemma 4.8.2 and (4.10) that Z Z m.n/a.3n/ .n/a.3n/ 5..n/ m/ .x C um/.s/ds E.xjB /.s/ds X.a.3n//. a.3n/ ma.3n/ 9.n/ (4.11) Since .n/ 100m, it follows from (4.9) and (4.11) that Z m.n/a.3n/ 5..n/ m/ 1 1 um.s/ds X.a.3n// X.a.3n// X.a.3n//. 0 9.n/ 2 20 Therefore, Z Z m.n/a.3n/ 1 1 .n/a.3n/ E B m um.s/ds X.a.3n C 1// .xj 100 /.s/ds. 0 30 30 0 It follows immediately that the inequality Z Z t mt E B .xj 100m /.s/ds 30 .s, um/ds (4.12) 0 0 holds for every t D .n/a.3n/ provided that 2.n/a.3n/ < a.3n C 1/ and .n/ 100m. Since the left-hand side of the inequality (4.12) is a piecewise-linear function of t and the right-hand side is a concave one, it follows that

1 E.xjB 100m / 30u . m m m

Since um ! 0asm !1, the assertion follows.

If .n/ D m for all n 2 N, we write Bm instead of B.

Proposition 4.8.5. Let E be a fully symmetric function space equipped with a Fatou norm. If x D .x/ 2 E is such that '.y/ '.x/ for every positive symmetric 1 functional ' on E and every 0 y x, then m mE.xjBm/ ! 0 in the norm topology of E.

Ã RZ R Proof. Consider the mappings m : C ! deﬁned as follows

1 Ã ./ :D k E.xjB /k . m m m E We want to verify the conditions of Proposition 4.8.3. 138 Chapter 4 Symmetric Functionals

Suppose that 0 .Letn 2 Z be such that .n/2a.3n/ < a.3n C 1/. It trivially follows that 02.n/a.3n/ < a.3n C 1/. Therefore, Z Z Z .n/a.3n/ a.3nC1/ a.3n/ E.xjB /.s/ds x.s/ds D 3=2 x.s/ds 0 Z 0 Z0 0.n/a.3n/ 0.n/a.3n/ 3=2 x.s/ds D 3=2 E.xjB0 /.s/ds. 0 0 It follows that Z Z t t E.xjB /.s/ds 3=2 E.xjB0 /.s/ds, t>0. (4.13) 0 0

This proves that Ãm is quasi-decreasing. E B E B If k # ,then .xj k / converges to .xj / almost everywhere. Since the norm on E is a Fatou norm, it follows that Ãm is order-continuous on the right. Hence, the condition (b) of Proposition 4.8.3 holds. Setting 0.n/ :D a.3n C 1/=a.3n/ for every n 2 Z, we satisfy the condition (b) of Proposition 4.8.3. By Proposition 4.8.4, the condition (c) of Proposition 4.8.3 also holds. The assertion now follows from Proposition 4.8.3.

Lemma 4.8.6. If x D .x/ 2 L1 C L1 and if Ci ,1 i k, arediscretesets,then

Xk E k C E C .xj[iD1 i / .xj i /. iD1

Proof. It is sufﬁcient to verify the inequality Z Z t Xk t E k C E C .xj[iD1 i /.s/ds .xj i /.s/ds 0 iD1 0

E k C E C only at the nodes of .xj[iD1 i /, that is at the nodes of .xj i / for every i. However, if t 2 Ci for some i,then Z Z t t E k C E C .xj[iD1 i /.s/ds D X.t/ D .xj i /.s/ds 0 0 and the result is shown.

Set

2 Am :Dfma.n/ :wheren 2 Z is such that m a.n/ < a.n C 1/g. Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 139

Lemma 4.8.7. Let E be a fully symmetric function space equipped with a Fatou norm. If x D .x/ 2 E is such that '.y/ '.x/ for every positive symmetric functional ' 1 on E and every 0 y x, then m mE.xjAm/ ! 0 in the norm topology of E. Proof. Let

2 Cm :Dfma.3n C 1/ :wheren 2 Z is such that m a.3n C 1/

Bm [ Cm [ Dm D Am.

Therefore, by Lemma 4.8.6, we have

E.xjAm/ E.xjBm/ C E.xjCm/ C E.xjDm/. (4.14) 1 By Proposition 4.8.5, we have m mE.xjBm/ ! 0asm !1. Similarly, apply- 1 ing Proposition 4.8.5 to the functions 2x=3and4x=9, we obtain that m mE.xjCm/! 1 0andm mE.xjDm/ ! 0asm !1. The assertion follows immediately.

Lemma 4.8.8. Let x D .x/ 2 .L1 C L1/.0, 1/. If x … L1.0, 1/, then, for every t>0 and every m 2 N, we have Z m4 t 2 4 3 X.t/ X.m t/C E.xjAm/.s/ds, (4.15) 3 2 0 R t where X.t/ :D 0 x.s/ds. Proof. For a given t>0, there exists n 2 Z such that t 2 Œa.n/, a.n C 1/.We consider all the cases for the relationships amongst a.n/, a.n C 1/ and a.n C 2/.If a.n C 1/>m2a.n/,thenm4t ma.n/ and, therefore,

Z 4 Z m t ma.n/ 2 E.xjAm/.s/ds E.xjAm/.s/ds D X.ma.n// X.t/. 0 0 3 If a.n C 1/ m2a.n/ and a.n C 2/>m2a.n C 1/,thenm4t ma.n C 1/ and, therefore, Z Z m4t ma.nC1/ E.xjAm/.s/ds E.xjAm/.s/ds D X.ma.n C 1// X.t/. 0 0 If a.n C 2/ m2a.n C 1/ and a.n C 1/ m2a.n/,then 3 3 X.m4t/ X.a.n C 2// D X.a.n C 1// X.t/ 2 2 and the assertion follows. 140 Chapter 4 Symmetric Functionals

The situation in the case where x 2 L1 is slightly more complicated.

Lemma 4.8.9. If x D .x/ 2 L1.0, 1/ or x 2 L1.0, 1/, then there exists constant C such that for every t>0 Z Z m4t m4t 2 4 3 E A X.t/ X.m t/C .xj m/.s/ds C C Œ0,1.s/ds, (4.16) 3 2 0 0 R t where X.t/ :D 0 x.s/ds.

Proof. Consider ﬁrst the case of the semi-axis. Fix n0 such that X.a.n0// 4=9X.1/.Foragivent 2 .0, a.n0//, there exists n 2 Z such that n

Setting C :D X.1/= minfa.n0/,1g, we obtain the assertion. The same argument applies in the case of the interval .0, 1/ by replacing X.1/ with X.1/.

Proof of Theorem 4.8.1 (a). Without loss of generality, x D .x/.Ifx … L1,thenby Lemma 4.8.8,

Z 4 Z Z t=m 2 t 3 t x.s/ds x.s/ds C E.xjAm/.s/ds, 8t>0 0 3 0 2 0 or, equivalently, 1 2 3 4 x x C E.xjA /. m4 m 3 2 m

1 Applying m m to both sides of the last display, we obtain 1 2 1 3 1 5 x x C E.xjA /. m5 m 3 m m 2 m m m

We now apply the norm kkE to both sides of the last display and let m !1.It follows from Lemma 4.8.7 that 1 2 1 lim kmxkE lim kmxkE . m!1 m 3 m!1 m This proves the assertion. Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different 141

If x 2 L1 and C are as in Lemma 4.8.9, then it follows from Lemma 4.8.9 that

Z 4 Z Z Z t=m 2 t 3 t t x.s/ds x.s/dsC E.xjAm/.s/ds CC Œ0,1.s/ds, 8t>0 0 3 0 2 0 0 or, equivalently,

1 2 3 4 x x C E.xjA / C C . m4 m 3 2 m .0,1/

1 Applying m m to both sides of the last display, we obtain 1 2 1 3 1 1 5 x x C E.xjA / C C . m5 m 3 m m 2 m m m m m .0,1/

We apply the norm kkE to both sides of the last display and let m !1. For every symmetric space E on the interval .0, 1/ and for every symmetric space E on the 1 semi-axis such that E 6 L1.0, 1/ we have m km.0,1/kE ! 0. It follows from Lemma 4.8.7 that 1 2 1 lim kmxkE lim kmxkE m!1 m 3 m!1 m and the assertion follows.

Proof of Theorem 4.8.1 (b). Consider the Banach space .E C L1/.0, 1/ equipped with the norm given by the formula kxkECL1 :Dk.x/.0,1/kE . Observe that E C L1 is a fully symmetric space with a Fatou norm. Suppose that the assertion of Theorem 4.8.1 (b) fails. That is,

1 lim km.x/kECL1 > 0. m!1 m

Applying Theorem 4.8.1 (a) to the space ECL1, we ﬁnd that there exists 0 y x and a positive symmetric functional ' 2 .E C L1/ such that '.y/ > '.x/. Obvi- ously, y 2 E and ' is a symmetric functional on E. Thus, we obtained a contradiction. This proves the assertion of Theorem 4.8.1 (b).

The proof of Theorem 4.8.1 (c) is identical to that of Theorem 4.8.1 (a) and is, therefore, omitted. 142 Chapter 4 Symmetric Functionals

4.9 Symmetric Functionals on Symmetric Operator Spaces

Sections 4.6–4.8 have provided results concerning the existence of symmetric functionals, fully symmetric functionals, and the distinction between the two, for symmetric and fully symmetric function spaces. We can use the lifting established in Sec- tion 4.4 to transfer these results to statements about the existence of symmetric functionals on symmetric operator spaces. Let M be a von Neumann algebra equipped with a faithful normal semiﬁnite trace . Theorem 4.9.1 below establishes criteria for when a nontrivial symmetric functional exists on a symmetric operator space E.M, / associated to a symmetric function space E. Theorem 4.9.2, which imposes the condition that M be an atomless or atomic semiﬁnite von Neumann algebra, is a stronger result than Theorem 4.9.1. Using the Calkin correspondence we can associate to every symmetric operator space a symmetric function space, and bijectively lift the symmetrical functionals on the symmetric function space to symmetric functionals on the symmetric operator space. The assertion of Theorem 4.1.3 in the introduction to this chapter follows from Theorem 4.9.2 by setting M D L.H / and recalling, from Section 2.7, that the set of continuous traces on a symmetrically normed ideal of compact operators is identical to the set of symmetric functionals on that ideal.

Theorem 4.9.1. Let E be a symmetric function (respectively, sequence) space with corresponding symmetric operator space

E.M, / :DfA 2 S.M, / : .A/ 2 Eg , kAkE :Dk.A/kE .

(respectively

E.M, / :DfA 2 S.M, / : E..A/jA/ 2 Eg , kAkE :DkE..A/jA/kE . /

Consider the following conditions. (a) There exist nontrivial positive singular symmetric functionals on E.M, /.

(b) There exist nontrivial singular fully symmetric functionals on E.M, /.

(d) If .1/ D1and if E.M, / 6 L1.M, /, then there exists an operator A 2 E.M, / such that

1 ˚m lim kA kE > 0. (4.17) m!1 m Section 4.9 Symmetric Functionals on Symmetric Operator Spaces 143

If .1/ D1and if E.M, / L1.M, /, then there exists an operator A 2 E.M, / such that

1 ˚m lim kA kECL1 > 0. (4.18) m!1 m If .1/ D 1, then there exists an operator A 2 E.M, / such that

1 ˚ m 0 lim kA k.ECL1/.M˝l1,˝ / > 0. (4.19) m!1 m 0 Here, is the semiﬁnite trace on l1 given by the sum, as in Example 2.2.3 (a). (i) The condition (d) implies (a). (ii) The condition (d) implies the equivalent conditions (a), (b) if E is fully symmetric. (iii) The condition (d) implies the equivalent conditions (a)–(c) if E is fully symmetric and equipped with a Fatou norm. The proof of Theorem 4.9.1 is contained in the proof of the next theorem since none of the analogous implications (that is, in one direction) in Theorem 4.9.2 require the condition that the von Neumann algebra be atomless or atomic. In the atomic case we assume that the trace of every atom is 1.

Theorem 4.9.2. Let M be an atomless or atomic von Neumann algebra equipped with a ﬁxed faithful normal semiﬁnite trace .LetE.M, / be a symmetric operator space. Consider the following conditions. (a) There exist nontrivial positive singular symmetric functionals on E.M, /. (b) There exist nontrivial singular fully symmetric functionals on E.M, /. (c) There exist positive symmetric but not fully symmetric functionals on E.M, /.

(d) If .1/ D1and if E.M, / 6 L1.M, /, then there exists an operator A 2 E.M, / such that

1 ˚m lim kA kE > 0. (4.20) m!1 m

If .1/ D1and if E.M, / L1.M, /, then there exists an operator A 2 E.M, / such that

1 ˚m lim kA kECL1 > 0. (4.21) m!1 m If .1/ D 1, then there exists an operator A 2 E.M, / such that

1 ˚ m 0 lim kA k.ECL1/.M˝l1,˝ / > 0. (4.22) m!1 m 0 Here, is the semiﬁnite trace on l1 given by the sum, as in Example 2.2.3 (a). 144 Chapter 4 Symmetric Functionals

(i) The conditions (a) and (d) are equivalent. (ii) The conditions (a), (b) and (d) are equivalent if E.M, / is fully symmetric. (iii) The conditions (a)–(d) are equivalent if E.M, /is fully symmetric and equipped with a Fatou norm.

Proof. The implications (b) ) (a) and (c) ) (a) are trivial. (a) ) (d) Let E.M, / be a symmetric operator space with a nontrivial singular symmetric functional '.LetA 2 E.M, /be an operator such that '.A/ ¤ 0. Without loss of generality, A 0. Let .1/ D1and let E.M, / 6 L1.M, /.Wehave

1 ˚m 1 ˚m j'.A/jD j'.A /jk'kE kA kE . m m Taking the limit as m !1, we obtain the required inequality (4.20). Let .1/ D1and let E.M, / L1.M, /. Note that this assumption prevents M from being atomic (indeed, otherwise E.M, / D L1.M, / does not support any singular symmetric functional). Hence, M is atomless. Since ' is a singular functional (that is, ' vanishes on .L1 \ L1/.M, /) and since 1 A AE , A , 1 2 .L \ L1/.M, /, 8m 2 ZC, A m 1 it follows that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˚m ˇ ˇ 1 ˇ 1 ˇ 1 ˇ j'.A/jDˇ' AEA , A , 1 ˇ D ˇ' AEA , A , 1 ˇ m m m ˚m 1 1 k'kE AEA , A , 1 m m E

1 ˚m k'kE kA kECL . m 1 Taking the limit as m !1, we obtain the required inequality (4.21). If .1/ D 1, then the proof of (4.22) is identical to that of (4.21) in the above paragraph (and is, therefore, omitted). (d) ) (a) Firstly, we assume that the algebra M is -ﬁnite. Without loss of generality, .1/ D 1. Let E.M, / be a symmetric operator space and let E be the corresponding symmetric function space. By the assumptions, there exists an element 1 x D .A/ 2 E such that m mx 6! 0inE. By Theorem 4.6.1 (c), there exists a positive singular symmetric functional 0 ¤ ' 2 E.LetL.'/ be the functional on E.M, / deﬁned in Theorem 4.4.1. Clearly, L.'/ is a nontrivial positive symmetric functional on E.M, /. Section 4.9 Symmetric Functionals on Symmetric Operator Spaces 145

The case when M is an infinite atomless von Neumann algebra can be treated in a similar manner. The only difference is that the reference to Theorem 4.6.1 (c) has to be replaced with the reference to Theorem 4.6.1 (b) or Theorem 4.6.1 (a). Let E.M, /be a symmetric operator space on an atomic von Neumann algebra M and let E be the corresponding symmetric sequence space. It follows from the assumptions that E.M, / ¤ L1.M, / or, equivalently, E ¤ l1. By the assumptions, there 1 exists an element x D .A/ 2 E such that m mx 6! 0inE.LetF be the symmetric function space constructed in Proposition 3.6.3. Since E ¤ l1, it follows that F 6 L1.0, 1/. Recall that the sequence space E is isometrically embedded into the 1 function space F .Wehavex 2 F and m mx 6! 0inF . By Theorem 4.6.1, there exists a positive symmetric functional 0 ' 2 F . The restriction of the functional ' to E is a nontrivial positive symmetric functional on E.LetL.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a nontrivial positive symmetric functional on E.M, /. (d) ) (b) The proof is very similar to that of the implication (d) ) (a) and is, therefore, omitted. The only difference is that the references to Theorem 4.6.1 have to be replaced with references to Theorem 4.7.1. (d) ) (c) Firstly, we assume that the algebra M is -finite. Without loss of generality, .1/ D 1. Let E.M, /be a symmetric operator space and let E be the corresponding symmetric function space. By assumption, there exists an element x D .A/ 2 E 1 such that m mx 6! 0inE. By Theorem 4.8.1, there exists a positive symmetric but not fully symmetric functional ' 2 E.LetL.'/ be the functional on E.M, /defined in Theorem 4.4.1. Clearly, L.'/ is a symmetric but not fully symmetric functional on E.M, /. The case when M is an infinite atomless von Neumann algebra can be treated in a similar manner. Let E.M, / be a symmetric operator space on an atomic von Neumann algebra M and let E be the corresponding symmetric sequence space. It follows from the assumptions that E.M, / ¤ L1.M, / or, equivalently, E ¤ l1. By assumption, 1 there exists an element x D .A/ 2 E such that m mx 6! 0inE.LetF be the symmetric function space constructed in Proposition 3.6.3. Since E ¤ l1, it follows that F 6 L1.0, 1/. Recall that the sequence space E is isometrically embedded into 1 the function space F .Wehavex 2 F and m mx 6! 0inF . By Theorem 4.8.1, there exists a positive symmetric functional ' 2 F andafunction0 y x such that '.y/ > '.x/.Setz D E..y/jf.n 1, ngn2N/. Clearly, z 2 E and '.z/ D '.y/ > '.x/. Hence, the restriction of the functional ' to E is a positive symmetric functional on E that is not fully symmetric. Let L.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a positive symmetric functional on E.M, / that is not fully symmetric.

Corollary 4.9.3. Let E be a symmetric ideal of L.H /. The following conditions are equivalent. 146 Chapter 4 Symmetric Functionals

(a) There exist nontrivial continuous traces on E.

(b) There exists an operator A 2 E such that

1 ˚m lim kA kE > 0. (4.23) m!1 m

Proof. If E ¤ L1, then the assertion follows from Theorem 4.9.2 and Lemma 2.7.4. For E D L1 the assertion is obvious.

4.10 How Large is the Set of Symmetric Functionals?

In the preceding section we fully characterized, in the atomless or atomic case, the symmetric operator spaces with nonempty sets of symmetric functionals. In this section, we examine the structure of the latter set when the symmetric operator space is equipped with a Fatou norm.

Theorem 4.10.1. Let M be an atomless or atomic von Neumann algebra equipped with a faithful normal semiﬁnite trace . Let E.M, / be a symmetric operator space equipped with a Fatou norm. One of the following mutually exclusive statements holds. (a) The space E.M, / does not admit a nontrivial symmetric functional.

(b) The space E.M, /admits a (up to a constant) unique nontrivial symmetric functional and it is a faithful normal semiﬁnite trace.

Proof. The set of Hermitian symmetric functionals on E.M, / is a Banach lattice (see Theorem 4.3.5). We need to show that its dimension is either 0 or 1 or 1.IfM is atomless, then the assertion follows from Theorem 4.10.5 (see below) and Theo- rem 4.4.1. If M D l1, then the assertion follows from Theorem 4.10.5 (see below) and Theorem 4.4.4. If M is atomic, then the assertion follows from Theorem 4.4.1.

We now proceed with the proof of Theorem 4.10.5 which is the key (commutative) ingredient in the proof of Theorem 4.10.1. If E is a symmetric function space equipped with a Fatou norm then the following lemma proves that the mapping on E,

1 x ! lim k.m.x//.0,1/kE m!1 m is never additive. Section 4.10 How Large is the Set of Symmetric Functionals? 147

Lemma 4.10.2. Let E D E.0, 1/ (or, E D E.0, 1/) be a symmetric function space equipped with a FatouP norm and let 0 x 2 E. For every n 2 N, there exists a n decomposition x D kD1 xk, such that xk 0, 1 k n, and such that 1 1 lim k.m.x//.0,1/kE D lim k.m.xk//.0,1/kE ,1 k n. m!1 m m!1 m Proof. Without loss of generality, x D .x/. Moreover, we can assume that supp.x/ .0, 1/.Thatis,x D .x/.0,1/. Fix m 2 N.Ifk !1, then we obtain

..x/ 1 1 / " ..x/ 1 / m . k , m / m .0, m / almost everywhere. By the deﬁnition of a Fatou norm, it follows that

k ..x/ 1 1 /k !k ..x/ 1 /k . m . k , m / E m .0, m / E For each m 2 N, select a number f.m/>m, such that 1 km..x/ 1 1 /kE 1 km..x/ 1 /kE . (4.24) . f.m/, m / m .0, m /

i i Set m0 D 1andletmi :D f .1/, i 2 N. Here, f :D f ı ıf (i times). It follows from the deﬁnition of miC1 that 1 k i ..x/ 1 1 /k 1 k i ..x/ 1 /k . (4.25) m . m , m / E m .0, m / E iC1 i mi i

Deﬁne the sets Ak ,1 k n, by setting [ 1 1 Ak :D , . miC1 mi iDkmodn

Set xk :D .x/Ak ,1 k n. It is clear that [ Ak D .0, 1/ 1kn P n and, therefore kD1 xk D .x/.0,1/ D x. If i D k mod n,then k.mi .xk//.0,1/kE kmi ..x/ 1 1 /kE . m , m iC1 i It now follows from (4.25) that 1 1 1 k.mi .xk//.0,1/kE 1 k.mi .x//.0,1/kE . mi mi mi 148 Chapter 4 Symmetric Functionals

Letting i D k mod n !1, we obtain mi !1and, therefore, 1 1 lim k.m.xk//.0,1/kE lim k.m.x//.0,1/kE . m!1 m m!1 m The converse inequality is obvious.

Lemma 4.10.3. Let E D E.0, 1/ 6 L1.0, 1/ be a symmetric function space equipped with a Fatou norm andP let 0 x 2 E be bounded. For every n 2 N, n there exists a decomposition x D kD1 xk such that xk 0, 1 k n, and such that 1 1 lim km.x/kE D lim km.xk/kE ,1 k n. m!1 m m!1 m Proof. Without loss of generality, x D .x/. Fix m>0. If k !1, then we obtain that

m..x/.0,k// " m.x/.

By the deﬁnition of a Fatou norm, it follows that

km..x/.0,k//kE !km.x/kE .

For each m 2 N, select a number f.m/>m, such that 1 k ..x/ /k 1 k .x/k . (4.26) m .0,f.m// E m m E

Set m0 D 0 and deﬁne the strictly increasing sequence mi , i 2 N, by setting mi :D f i1.1/. Here, f i :D f ııf (i times). We have 1 kmi ..x/.0,miC1 //kE 1 kmi .x/kE , i 0. (4.27) mi

Since E 6 L1.0, 1/, it follows that the fundamental function of E satisﬁes the condition .t/ D o.t/ as t !1. Passing to a subsequence (if needed), we preserve (4.27) and obtain that

i .mi miC1/ 2 miC1, i 0. (4.28)

For 1 k n, deﬁne the set Ak by setting [ Ak :D .mi , miC2/. iD2k mod 2n

Set xPk :D .x/Ak ,1 k n. It is clear that [1knAk D .0, 1/ and, therefore, n x D kD1 xk. Section 4.10 How Large is the Set of Symmetric Functionals? 149

For every 1 k n and every i D 2k mod n,wehave

kmiC1 .xk/kE kmiC1 ..x/.mi ,miC2//kE

kmiC1 ..x/.0,miC2 //kE kmiC1 ..x/.0,mi //kE

kmiC1 ..x/.0,miC2 //kE kxk1 .mi miC1/.

It follows from (4.27) and (4.28) that 1 i kmiC1 .xk/kE 1 kmiC1 .x/kE 2 miC1 kxk1. miC1

Letting i !1, we obtain mi !1and, therefore, 1 1 lim km.xk/kE lim km.x/kE . m!1 m m!1 m The converse inequality is obvious.

Lemma 4.10.4. Let E D E.0, 1/ 6 L1.0, 1/ be a symmetric function space and let 0 x 2 E \ L1. We have 1 1 lim km.x/kE D lim k.m.x//.0,1/kE . m!1 m m!1 m Proof. Let be the (concave majorant of the) fundamental function of E. It follows from Theorem II.5.5 of [139] that

kzkE kzkM , 8z 2 E.

It now follows that 1 k.m.x//.1,1/kE k , x .0,1/ C .m.x//.1,1/kE m 1 k , x .0,1/ C .m.x//.1,1/kM m Z 1 1 D .1/ , x C m 0.ms/.s, x/ds. m 1=m

Therefore, Z 1 1 1 1 0 k.m.x//.1,1/kE , x .1/ C .ms/.s, x/ds. m m m 1=m

We have 0 a.e. .ms/.s, x/.1=m,1/.s/ ! 0 150 Chapter 4 Symmetric Functionals and 0 .ms/.s, x/.1=m,1/.s/ .s, x/.

It now follows from the Dominated Convergence Theorem that Z 1 0.ms/.s, x/ds ! 0. 1=m

1 Since x 2 L1, it follows that .1=m, x/ D o.m / and, therefore, 1 k. .x// 1 k ! 0. m m .1, / E The assertion now follows from the triangle inequality.

Theorem 4.10.5. Let E be a symmetric function space equipped with a Fatou norm. One of the following mutually exclusive statements holds. (a) The space E does not admit a nontrivial symmetric functional.

(b) The space E admits a (up to a constant) unique nontrivial symmetric functional, that is, an integral.

Proof. We consider only the case E D E.0, 1/ 6 L1.0, 1/ (this automatically excludes (b)). All the remaining cases can be treated identically (using Lemma 4.10.2 instead of the combination of Lemma 4.10.2, Lemma 4.10.3 and Lemma 4.10.4). Suppose that there exists a symmetric functional on E. This excludes (a). Now, we show that (c) holds. By Theorem 4.2.2 there exists a positive symmetric functional on E. By Theorem 4.9.2 (or Theorem 4.6.1), there exists an element x D .x/ 2 E such that 1 lim km.x/kE > 0. m!1 m It follows that either 1 lim km..x/.0,1//kE > 0 m!1 m or 1 lim km.minf.x/, .1/g/kE > 0. m!1 m

Without loss of generality we have either x D minf.x/, .1/g or x D .x/.0,1/. Section 4.10 How Large is the Set of Symmetric Functionals? 151

Consider theP ﬁrst case. By Lemma 4.10.3, there exist positive elements xk,1 k n n, such that kD1 xk D x and 1 1 lim km.xk/kE D lim km.x/kE > 0. m!1 m m!1 m

Consider the secondP case. By Lemma 4.10.2, there exist positive elements xk,1 n k n, such that kD1 xk D x and 1 1 lim k.m.xk//.0,1/kE D lim k.m.x//.0,1/kE > 0. m!1 m m!1 m By Lemma 4.10.4, we have

1 1 lim km.xk/kE D lim km.x/kE > 0. m!1 m m!1 m By Theorem 4.7.1, for every 1 k n, there exists a singular fully symmetric functional 'k such that k'kkE 1and 1 1 'k.xk/ D lim km.xk/kE D lim km.x/kE > 0. m!1 m m!1 m

Since 'k is a positive functional, we have 1 1 ' .x / ' .x/ D ' . .x// k .x/k . k k k m k m m m E Since m is arbitrarily large, we have

1 'k.x/ D 'k.xk/ D lim km.x/kE . m!1 m

Therefore, 'k.xl / D 0 for every l ¤ k. It clearly follows that functionals 'k,1 k n, are linearly independent. Thus, for an arbitrarily large n 1, there exist n linearly independent fully symmetric functionals on E. Hence, the linear space of all fully symmetric functionals on E cannot be ﬁnite-dimensional. Therefore, it is inﬁnite-dimensional. Since it is a closed subspace of a Banach space E, then it is a Banach space itself and the assertion follows. 152 Chapter 4 Symmetric Functionals

4.11 Notes

Symmetric Functionals The concept of a symmetric functional (as a positive continuous functional which takes the same value on those positive elements with the same distribution function) was introduced in 1998 in the paper [67]. This concept bears a strong resemblance to the Calkin correspondence between symmetrically normed ideals of compact operators and symmetric sequence spaces. Indeed,if we study whether symmetric (quasi-)norms canbe lifted from commutative sequence and/or function spaces to their noncommutative counterparts, then it is natural also to ask whether linear symmetric functionals are also lifted. However, it should be noted that in the literature devoted to the study of singular functionals on Lorentz function spaces, as discussed in the end notes to Chapter 2, the concept of a symmetric functional was not introduced. Our first contact with such an object came from noncommutative geometry, from Alain Connes’ use of Dixmier traces and the subsequent exposure of Dixmier traces to a wide audience. The same term “symmetric functionals” was used by Figiel and Kalton [92] who studied a similar object (but without any reference to continuity or positivity). We refer to [92] for additional historical comments, and also for potential connections between symmetric functionals and analytic functions. Theorem 4.4.1 provides a natural bijection between the set of all symmetric functionals on a symmetric operator space E.M, / and the set of all symmetric functionals on the corresponding symmetric function space E, observed first for the case of the set of fully symmetric functionals in [67]. With regard to Theorem 4.6.2, which states that a symmetric functional decompose into normal and singular symmetric functionals, it was already established in [67, Proposition 2.2] that if E was a symmetric function space and 0 <'2 E was a nontrivialR normal and 1 symmetric functional, then necessarily E L1.0, 1/ and '.f / D ˛ 0 f.s/ds for all f 2 E and some constant ˛>0. Question 4.1.2 was suggested in [65,103,104,127]. Question 4.1.2 (c) was answered in the affirmative in [128] for the special case of a Lorentz ideal of compact operators. It should be pointed out that the method used in [128] cannot be extended to an arbitrary Lorentz ideal M .M, /and, furthermore, cannot be extended to a general symmetric operator space. The exposition in this chapter closely follows that of [239]. Our strategy in this chapter is based on the approach from the recent papers [240] and [130] (see also [20]), where condition(4.1) was connectedto the geometryof E.M, /.The condition (4.1) is easy to verify in concrete situations. For example, the main result of [128] follows immediately from Theorem 4.1.3. Some examples of symmetric spaces, which are not Lorentz spaces, that admit symmetric traces were presented in [65]. These results are also an immediate corollary of Theorem 4.1.3. Theorem 4.10.1 is a response to questions of V. Kaftal and G. Weiss [119,121] concerning the codimension of the commutator subspace (see the next chapter). They conjectured that it is always 0, or 1, or 1. The result in Theorem 4.10.1 confirms this conjecture for symmetric operator spaces, although in a rephrased form, since it confirms that the codimension of the closure of the commutator subspace, as studied in the next chapter, is 0, or 1, or 1. Chapter 5 Commutator Subspace

5.1 Introduction

In this chapter we look at the Lidskii formula for continuous traces on symmetrically normed ideals of compact operators. Let H be a separable Hilbert space. In this chapter we consider only the atomic factor M D L.H / of all bounded operators on H . Recall from Section 2.5 that symmetrically normed ideals are all the symmetric operator spaces for L.H /, and, from Sec- tion 2.7, that the set of continuous traces on a symmetrically normed ideal is the set of symmetric functionals on that ideal. According to Theorem 4.4.1, if E :D E.L.H /,Tr/ is a symmetrically normed ideal of compact operators and ' 2 E is a continuoustrace, then there exists a unique symmetric functional f 2 E on the Calkin sequence space E such that '.A/ D f ı .A/,0 A 2 E, where .A/ is the sequence of singular values of the compact operator A. This iden- tiﬁcation can be extended to eigenvalues. Throughout this chapter .A/ denotes an eigenvalues sequence of a compact operator A (see Deﬁnition 1.1.10).

Theorem 5.1.1 (Lidskii formula for continuoustraces). If E is a symmetrically normed ideal of compact operators with Calkin sequence space E, and ' 2 E is a continuous trace with corresponding symmetric functional f 2 E, then any eigenvalue sequence .A/ 2 E for A 2 E and

'.A/ D f ı .A/, A 2 E.

This result is not trivial, as the technicalities about quasi-nilpotent operators in Sec- tion 5.5 demonstrate. Given the Lidskii formula, the bijective correspondence between continuous traces on a symmetrically normed ideal and symmetric functionals on the Calkin sequence space is unequivocal:

' D f ı , f D ' ı diag, where diag is the diagonal operator for any orthonormal basis of H . Part III will examine the consequences of the Lidskii formula, and some variants of it, for Dixmier traces. 154 Chapter 5 Commutator Subspace

To obtain the Lidskii formula, we study the commutator subspace Com.E/ of a symmetrically normed ideal of compact operators E, and its closure Com.E/ in the norm topology of E.

Deﬁnition 5.1.2. Let E be a symmetrically normed ideal of compact operators. The subspace Com.E/ :D Lin.ŒA, B : A 2 E, B 2 L.H // is called the commutator subspace of E. Its closure in the norm topology of E

Com.E/ :D Lin.ŒA, B : A 2 E, B 2 L.H // is called the closed commutator subspace. The commutator subspace is the common kernel of all traces (meaning here all unitarily invariant linear functionals on E), and the closed commutator subspace is the common kernel of all continuous traces on E (this latter fact is shown in Section 5.3). That is, ' 2 E is a continuous trace if and only if ' vanishes on Com.E/. Deﬁne the Cesàro operator

C : l1 ! l1 by Xn 1 1 1 C.x/ :D x , x Dfx g 2 l1. 1 C n j j j D0 j D0 nD0

The commutator subspace and the invariance of the Calkin space to the Cesàro operator are related by the following noncommutative analog of the Figiel–Kalton theorem. It is the main result of this chapter. The Lidskii formula will follow from this result.

Theorem 5.1.3. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. If A 2 E, then .A/ 2 E and (a) A 2 Com.E/ if and only if C..A// 2 E. (b) A 2 Com.E/ if and only if 1 1 1 C..A// ! 0 log.m/ m m in the norm topology of E. Here, .A/ denotes an eigenvalue sequence of A. Part (a) is a known result from the combined work of Kalton and Dykema, Figiel, Weiss and Wodzicki. Part (b) is new. Section 5.6 discusses some corollaries of the Section 5.2 Normal Operators in the Commutator Subspace 155 spectral identiﬁcation in Theorem 5.1.3, including the proof of Theorem 5.1.1. Theo- rem 5.1.3 will also be pivotal to the applications in Part IV.

5.2 Normal Operators in the Commutator Subspace

Let .E, kkE / be a symmetrically normed ideal of compact operators with corresponding Calkin sequence space E. Theorem 5.1.3 is proved by two steps. The ﬁrst step is lifting appropriate versions of the Figiel–Kalton theorem (one version for the commutator subspace and one version for the closed commutator subspace) from the sequence space E to the ideal of compact operators E. The lifting will provide the result of Theorem 5.1.3 for normal compact operators in E. The second step is to use Ringrose’s result (Theorem 1.1.22 in Chapter 1): if A is a compact operator then there exists a normal compact operator N and a compact quasi-nilpotent operator Q such that

A D N C Q and A and N can have the same eigenvalue sequence, .A/ D .N /. We will show that if A belongs to a symmetrically normed ideal of compact operators E,thenN (and hence Q) also belongs to E. Theorem 5.1.3 for an arbitrary A 2 E will follow from the ﬁrst step when we show, in Section 5.5, that the quasi-nilpotent operator Q 2 Com.E/. The two steps above are the master plan for proving Theorem 5.1.3. In this section we show the statement of Theorem 5.1.3 (a) for normal operators. For the background of this result, which originated in [80], see the end notes of this chapter.

Theorem 5.2.1. Let A 2 E be a normal operator and let .A/ 2 E be an eigenvalue sequence of A. We have A 2 Com.E/ if and only if C.A/2 E. We prove Theorem 5.2.1 using the following lemmas.

Lemma 5.2.2. Let E be a symmetrically normed ideal of compact operators. For all positive operators A1, A2 2 E such that .A1/ D .A2/, we have A1 A2 2 Com.E/. In particular, 2A A ˚ A 2 Com.E/.

Proof.LetU be a partial isometry such that U U D EA1 .0, 1/, UU D EA2 .0, 1/ and UA1U D A2. It follows that

A1 A2 D A1 UA1U D U UA1 UA1U D ŒU , UA1 2 Com.E/.

In particular,

2A A ˚ A D .A A ˚ 0/ C .A 0 ˚ A/ 2 Com.E/ C Com.E/ D Com.E/. 156 Chapter 5 Commutator Subspace

The next elementary lemma is stated (but not proved) in [178]. P Rn n1 Lemma 5.2.3. Let 2 be such that kD0 k D 0. There exists a permutation of the set f0, 1, :::, n 1g such that ˇ ˇ ˇ Xk ˇ sup ˇ .i/ˇ sup jkj. 0kn1 0kn1 iD0

Proof. Without loss of generality, all the values k,0 k n 1, are distinct. We may assume that .0, C/ .0, /. There exists a unique number .0/ such that .0/ D .0, C/.Setn0 D 1. We use induction to deﬁne nk andP.k/,0 k k n 1. Suppose that ni and .i/,0 i k, are already deﬁned.P If iD1 .i/ < 0, k then set .kC1/ :D .nk , C/ and nkC1 :D nk C 1. If iD1 .i/ 0, then set .kC1/ :D.k C 1 nk, C/ and nkC1 :D nk.

Lemma 5.2.4. Let A 2 Mn.C/ be a self-adjoint operator such that Tr.A/ D 0. There exists an operator B 2 Mn.C/ and a partial isometry U 2 Mn.C/ such that A D ŒU , B and kBk1 kAk1.

Proof. Let k,0 k n 1, be the eigenvalues of the operator A.Usingthe permutation constructed in Lemma 5.2.3, we can assume that ˇ ˇ ˇ Xk ˇ ˇ i ˇ kAk1,0 k n 1. iD0

Select a basis in which A corresponds to the diagonal matrixP diag.0, :::, n1/.Let n1 Ei,j ,0 i, j n 1, be matrix units. Set U :D iD1 Ei1,i .WehaveA D ŒU , U D with

D D diag.0, 0 C 1, :::, 0 CCn1/.

The claim follows by setting B :D U D.

Lemma 5.2.5. For every positive operator A 2 E and every positive operator B such n nC1 that .B/ D E..A/jf0g[fŒ2 ,2 /gn0/, we have A B 2 Com.E/.

Proof.Letek 2 H , k 0, be such that Aek D .k, A/ek . Without loss we may assume that Bek D .k, B/ek.Letpn be the projection on the linear span of ek, k 2 Œ2n,2nC1/. It is clear that

n Tr.pn.A B/pn/ D 0, kpn.A B/pnk1 .2 , A/. Section 5.2 Normal Operators in the Commutator Subspace 157

By Lemma 5.2.4, there exist operators Cn D pnCnpn and partial isometries Un D pnUnpn such that n pn.A B/pn D ŒUn, Cn, kCnk1 .2 , A/. The series X1 X1 C D Cn, U D Un nD0 nD0 converge strongly. Clearly, U is a partial isometry. We have M1 X1 n .C / D .Cn/ .2 , A/Œ2n,2nC1/ 2.A/ nD0 nD0 and, therefore, C 2 E. Hence, X X A B D pn.A B/pn D ŒUn, Cn D ŒU , C2 Com.E/. n0 n0 Recall that for any two-sided ideal E,ifA 2 E then

A D

A D AC A for positive operators AC, A 2 E. Lemma 5.2.6. Let E be a symmetrically normed ideal of compact operators. If the self-adjoint operator A D A 2 E is such that C..AC/ .A// 2 E, then A 2 Com.E/.

Proof. Fix an orthonormal basis ek, k 0. Deﬁne positive operators B1 and B2 by setting B1ek D .k, B1/ek and B2ek D .k, B2/ek,where n nC1 .B1/ D E..AC/jf0g[fŒ2 ,2 /gn0/ and n nC1 .B2/ D E..A/jf0g[fŒ2 ,2 /gn0/.

If 2n k<2nC1,then ˇ ˇ ˇ ˇ ˇ Xk ˇ ˇ Xk ˇ ˇ ..l, B1/ .l, AC// ˇ D ˇ ..l, B1/ .l, AC// ˇ lD0 lD2n n n 2.k 2 C 1/.2 , AC/. 158 Chapter 5 Commutator Subspace

Therefore,

jC.B1/ C.AC/j22.AC/, jC.B2/ C.A/j22.A/.

It now follows from the hypothesis in the statement that

jC..B1/ .B2//jjC..B1/ .AC//jCjC..B2/ .A//j

CjC..AC/ .A//j2E.

Deﬁne a sequence z by setting

Œ log2.kC1/ zk :D .C..B1/ .B2///.2 1/, k 0.

It is clear that

jzkjj.C..B1/ .B2///.k/jC.k, B1/ C .k, B2/ and, therefore, z 2 E. On the other hand, we have

z D .z0, z1, z3, z3, z7, z7, z7, z7, z15, :::/ and, therefore,

2z 2z D .z0,2z1 z0,2z3 z1,2z3 z1, :::/D .B1/ .B2/.

Deﬁne self-adjoint operators D1, D2 2 E by setting

D1 :D diag.z0, z1, :::/, D2 :D diag.z0, z0, z1, z1, :::/.

Clearly, D2 is equimeasurable with D1 ˚ D1. By Lemma 5.2.2, we have

B1 B2 D 2D1 D2 2 Com.E/.

It follows from Lemma 5.2.2 that

A D AC A D .AC B1/ .A B2/ C .B1 B2/ 2 Com.E/.

Lemma 5.2.7. Let A be a self-adjoint compact operator. We have

jC..A/ .AC/ C .A//j2.A/.

Proof. It is clear that

n n n f.k, A/gkD0 f.k, AC/gkD0 [f.k, A/gkD0. Section 5.2 Normal Operators in the Commutator Subspace 159

On the other hand, the set

n n n .f.k, AC/gkD0 [f.k, A/gkD0/nf.k, A/gkD0 f : jjj.n, A/jg is of cardinality less than 2.n C 1/. It follows that ˇ ˇ ˇ Xn Xn ˇ ˇ ..k, AC/ .k, A// .k, A/ˇ 2.n C 1/j.n, A/j. kD0 kD0 The assertion follows immediately.

Lemma 5.2.8. If A1, A2 2 E are self-adjoint operators, then

C.A1/ C C.A2/ C.A1 C A2/ 2 E.

Proof. Let A1, A2 2 E be positive. It follows from Theorem 3.3.3 that

C.A1 C A2/ C.A1/ C C.A2/. On the other hand, it follows from Theorem 3.3.4 that Z Z t 2t ..s, A1/ C .s, A2// ds .s, A1 C A2/ds 0 Z0 t .s, A1 C A2/ds C t.t, A1 C A2/. 0

Therefore, C..A1/ C .A2// C.A1 C A2/ C .A1 C A2/. Hence,

C.A1 C A2/ 2 C.A1/ C C.A2/ C E. For every sequence a,setfag to be the set of all sequences of the form a C b, b 2 E. It follows that the mapping : A !fC.A/g is additive on the positive cone of E. Therefore, admits a unique linear extension to the set of all self-adjoint operators in E, given by the formula

: A !fC..AC/ .A//g, A D A 2 E. However, it follows from Lemma 5.2.7 that

C..AC/ .A/ .A// 2 E for every self-adjoint operator A 2 E. Equivalently, .A/ DfC.A/g for every self- adjoint operator A 2 E.Since is linear, the result is shown.

The following lemma is a simpliﬁed version of Proposition 5.7 from [80]. 160 Chapter 5 Commutator Subspace

Lemma 5.2.9. Let A 2 E and B 2 L.H / be self-adjoint operators. We have

jC..ŒA, B//j4kBk1.A/ C .ŒA, B/.

Proof. Fix n 2 N and select projections p, q such that Tr.p/ D Tr.q/ D n and

kŒA, B.1 p/k1 D .n, ŒA, B/, kA.1 q/k1 D .n, A/. Setting r :D q _ p, we obtain

rŒA, Br D ŒrAr, rBr C rA.1 r/Br rB.1 r/Ar and, therefore,

Tr.rŒA, Br/ D Tr.rA.1 r/Br rB.1 r/Ar/. Hence,

jTr.rŒA, Br/j2kA.1 r/k1 kBk1 Tr.r/ 4n.n, A/kBk1. On the other hand, we have rŒA, Br D pŒA, Bp C .r p/ŒA, B.r p/ C pŒA, B.r p/ C .r p/ŒA, Bp and, therefore,

Tr.pŒA, Bp/ D Tr.rŒA, Br/ Tr..r p/ŒA, B.r p//.

We have 1 p r p and, therefore,

jTr..r p/ŒA, B.r p//jDjTr..r p/.1 p/ŒA, B.1 p/.r p//j

k.1 p/ŒA, B.1 p/k1 Tr.r p/ n.n, ŒA, B/. It follows that ˇ ˇ ˇ Xn 1 ˇ ˇ .k, ŒA, B/ˇ DjTr.pŒA, Bp/j4n.n, A/kBk1 C n.n, ŒA, B/, kD0 which provides the claim.

Lemma 5.2.10. For every compact normal operator A, we have

jC..A/ .

Proof. Fix n 0. Since A is a normal operator, it follows that

n .A/ \f : jj >.n, A/gf.k, A/gkD0. Section 5.2 Normal Operators in the Commutator Subspace 161

On the other hand, the set

n f.k, A/gkD0n..A/ \f : jj >.n, A/g/ has cardinality less than nC1 and is contained in the disk fjj.n, A/g. Therefore, ˇ ˇ ˇ Xn X ˇ ˇ .k, A/ ˇ .n C 1/.n, A/. (5.1) kD0 2.A/,jj>.n,A/

Since A is a normal operator, it follows that <.A/ D ..n,A/

We have ˇ ˇ ˇ X X ˇ X ˇ < <ˇ j<j 2.A/,j<j>.n,A/ 2.A/,Xjj>.n,A/ 2.A/,j<j.n,A/.n,A/

It now follows from (5.2) that ˇ ˇ ˇ Xn X ˇ ˇ .k, .n,A/

Combining (5.3) and (5.4), we obtain ˇ ˇ ˇ Xn X ˇ ˇ ..k, .n,A/ 162 Chapter 5 Commutator Subspace

It now follows from (5.1) that ˇ ˇ ˇ Xn ˇ ˇ ..k, A/ .k,

Proof of Theorem 5.2.1. Let A 2 Com.E/. It is clear that

Taking real parts, we obtain

It follows from Lemma 5.2.9 that

C.iŒ

It follows from Lemma 5.2.8 that C.

5.3 Normal Operators in the Closed Commutator Subspace

In this section we prove a version of the Figiel–Kalton theorem for symmetric functionals on a symmetric sequence space E. Recall the deﬁnitions of the sets DE and ZE , which were introduced at the beginning of Section 4.5,

DE :D Lin.fx 2 E : x D .x/g/ and

ZE :D Lin.fx1 x2 :0 x1.x2 2 E, .x1/ D .x2/g/. Section 5.3 Normal Operators in the Closed Commutator Subspace 163

Theorem 5.3.1 (closed version of the Figiel–Kalton Theorem). Let E be a symmetric sequence space and let x 2 DE . Then '.x/ D 0 for every symmetric functional ' on E if and only if 1 1 1 Cx ! 0 log.m/ m m in the norm topology of E. The proof is given below. As in the last section, we lift the result from a symmetric sequence space E to a statement about normal operators in the closed commutator subspace of a symmetrically normed ideal E. This will prove Theorem 5.1.3 (b) for normal operators in E, which is the main result of this section.

Theorem 5.3.2. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. Let A 2 E be a normal operator and let .A/ 2 E be an eigenvalue sequence of A. It follows that A 2 Com.E/ if and only if 1 1 1 C..A// ! 0 log.m/ m m in the norm topology of E. We now prove Theorems 5.3.1 and 5.3.2. The next result shows that Com.E/ is the common kernel for all symmetric functionals on E.

Lemma 5.3.3. Let E be a symmetrically normed ideal of compact operators. Then A 2 Com.E/ if and only if '.A/ D 0 for every symmetric functional ' 2 E. Proof. The only if direction is simple. Let ' 2 E be a symmetric functional. By Lemma 2.7.4 we have that ' is a trace and hence '.A/ D 0forA 2 Com.E/.Since' is continuous '.A/ D 0forA 2 Com.E/. The converse is more involved. Consider the convex functional p : E ! R deﬁned by setting p.A/ :D inffkA BkE : B 2 Com.E/g, A 2 E.

Using the Hahn–Banach theorem, we construct a linear functional ' p such that '.A/ D p.A/.Since' p, it follows that ' is a bounded functional on E. Clearly, the constructed functional ' vanishes on Com.E/. Recall that E consists of compact operators. If A1 and A2 are positive compact operators such that .A1/ D .A2/, then there exists a partial isometry U 2 L.H / such that U A2U D A1 and UU A2 D A2. It follows that

'.A2/ D '.UU A2/ D '.U.U A2// D '..U A2/U / D '.U A2U/D '.A1/.

Therefore, ' is a symmetric functional on E. 164 Chapter 5 Commutator Subspace

By assumption, '.A/ D 0. It follows that p.A/ D 0 or, equivalently, A 2 Com.E/.

Corollary 5.3.4. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E, and let A D A 2 E be a self-adjoint operator. It follows that A 2 Com.E/ if and only if '..AC/.A// D 0 for every symmetric functional ' 2 E.

Proof. From Lemma 5.3.3, we have that A 2 Com.E/ if and only if '.A/ D 0for every symmetric functional ' 2 E. Applying Theorem 4.2.2, we obtain that A 2 Com.E/ if and only if '.A/ D 0 for every positive symmetric functional ' 2 E.Ap- plying Theorem 4.4.1, we obtain that A 2 Com.E/ if and only if '..AC/.A// D 0 for every positive symmetric functional ' 2 E. Applying Theorem 4.2.2, we obtain that A 2 Com.E/ if and only if '..AC/ .A// D 0 for every symmetric functional ' 2 E.

Lemma 5.3.5. Let E be a symmetric sequence space. The operator D deﬁned by setting 1 1 Xn .Dx/ :D x C x , x 2 E n n C 1 n .n C 1/2 k kD0 is a bounded operator from E into the separable part E0 of E.

Proof. Let be (the concave majorant of) the fundamental function of E. It is known (see e.g. [139]) that M .t/ E Mt= .t/. Moreover,

kxk kxk kxk . Mt= .t/ E M .t/

Evidently, for every x 2 E,wehave

1 Xn 1 n C 1 j.Dx/ jjx jC .k, x/ jx jC kxk . n n .n C 1/2 n .n C 1/2 Mt= .t/ .n C 1/ kD0

Therefore, 1 kDxkE kxkE CkxkMt= .t/ .n C 1/ .n C 1/ n0 E 1 kxkE CkxkE . .n C 1/ .n C 1/ n0 M .t/ Section 5.3 Normal Operators in the Closed Commutator Subspace 165

We have X X 0 1 .n C 1/ .n/ .n/ D 1 C .n C 1/ .n C 1/ .n C 1/ .n C 1/ .n C 1/ .n/ n0 M .t/ n0 n1 Z 0.1/ 1 d log. .s// 1 C C 2 .1/ 1 s C 1 ˇ Z ˇ1 1 3 log. .s//ˇ log. .s// C ˇ C 2 ds. 2 s C 1 1 1 .s C 1/ Since the latter integral converges, the assertion follows immediately.

Lemma 5.3.6. For every x D .x/ 2 l1, we have 1 1 2 2 1 Cx C x C Dx C Dx (5.5) log.m/ m m log.m/ m log.m/ m and 1 1 1 2 x C 1 CxC Dx. (5.6) m m log.m/ m m log.m/ Here, D is the operator deﬁned in Lemma 5.3.5.

Proof. For every n 0 and every t 2 .n, n C 1/,wehave ˇ Z ˇ ˇ Z Z ˇ ˇ t ˇ ˇ t nC1 ˇ ˇ1 ˇ ˇ1 1 ˇ ˇ x.s/ds .Cx/nˇ D ˇ x.s/ds x.s/dsˇ 2.Dx/n. t 0 t 0 n C 1 0 It follows that ˇ Z ˇ ˇ t ˇ ˇ1 1 ˇ 1 ˇ x.s/ds 1 m Cx ˇ 2 Dx C mDx . t t=m m n m n Note that, for every 0 ma b,wehave Z Z Z Z b 1 t b minfms,bg dt x.s/dsdt D x.s/ds ma t t=m a maxfma,sg t Z b minfms, bg D x.s/log ds. a maxfma, sg

The integrand does not exceed x.s/log.m/ and, therefore, Z Z Z b 1 b b 1 1 m Cx .s/ds log.m/ x.s/ds C 2 Dx C mDx ds. ma m a ma m 166 Chapter 5 Commutator Subspace

Hence,

Z Xb 1 1 b 1 1 m Cx D 1 m Cx .s/ds m ma m kDma k Z Z b b 1 log.m/ x.s/ds C 2 Dx C mDx ds a a m 1 Xb 1 D log.m/x C Dx C Dx . k m m kDa k

The inequality (5.5) follows immediately. Proof of the inequality (5.6) is similar.

Lemma 5.3.7. Let E be a symmetric sequence space and let x 2 DE . It follows that 1 1 '.x/ D ' 1 Cx log.m/ m m for every symmetric functional ' 2 E.

Proof. If E D l1, then every symmetric functional is a summation and the assertion follows immediately. Suppose that E ¤ l1. It is sufﬁcient to prove the assertion for x D .x/. By Theorem 4.2.2, it is sufﬁcient to prove the assertion for a positive symmetric functional ' 2 E. It follows from Lemma 5.3.6 and Lemma 4.4.3 that 1 1 4 ' 1 Cx '.x/ C '.Dx/. log.m/ m m log.m/

By Lemma 5.3.5, Dx belongs to a separable part of E.SinceE ¤ l1, it follows that ' is a singular functional. Therefore, '.Dx/ D 0. It follows that 1 1 ' 1 Cx '.x/. log.m/ m m

Proof of the converse inequality is similar.

Proof of Theorem 5.3.1. Suppose ﬁrst that '.x/ D 0 for every symmetric functional ' 2 E, in particular for every positive symmetric functional ' on E.Set 1 1 p D lim sup 1 m Cx , 8x 2 DE . m!1 log.m/ m E Section 5.3 Normal Operators in the Closed Commutator Subspace 167

If x 2 ZE \ DE , then it follows from Theorem 4.5.1 that Cx 2 E. Hence, 1 1 m Cx 2kCxkE. m E

Therefore, p vanishes on ZE \ DE . Applying the argument of Lemma 4.6.5, we obtain a convex functional p : E ! R which vanishes on ZE . By the Hahn–Banach theorem, there exists a linear functional ' on E such that ' p and '.x/ D p.x/.Sincep.z/ D 0 for every z 2 ZE ,it follows that '.z/ D 0 for every z 2 ZE . Therefore, ' is a symmetric functional. By assumption, '.x/ D 0 and, therefore, p.x/ D 0. The assertion follows immediately. Conversely, suppose that 1 1 1 Cx ! 0. log.m/ m m

It follows from Lemma 5.3.7 that 1 1 '.x/ D ' 1 Cx ! 0. log.m/ m m

Proof of Theorem 5.3.2. Assume initially that A is self-adjoint. It follows from Corol- lary 5.3.4 and Lemma 5.3.1 that A 2 Com.E/ if and only if 1 1 1 C..AC/ .A// ! 0. log.m/ m m

By Lemma 5.2.7, we have

C..AC/ .A// C..A// 2 E.

Since 1 1 1 z ! 0, 8z 2 E, log.m/ m m the assertion follows immediately. Now assume that A is normal. It is clear that A 2 Com.E/ if and only if

By Lemma 5.2.10, we have

C..

Since 1 1 1 z ! 0, 8z 2 E, log.m/ m m the assertion follows immediately.

5.4 Subharmonic Functions on Matrix Algebras

In the last two sections we proved Theorem 5.1.3 for normal operators in a symmetrically normed ideal E of compact operators. To complete the proof of Theorem 5.1.3 requires that we show that Q 2 Com.E/ for a quasi-nilpotent operator Q 2 E.To show this result is technically quite difﬁcult. We follow the technique of Kalton and Dykema, based upon Kalton’s lifting of subharmonic functions to matrix algebras. We begin by recalling basic deﬁnitions.

A smooth function u : C ! R is called harmonic if u D 0. Equivalently, Z 1 2 u.z/ D u.z C eir/d 2 0 for every z 2 C and every r 2 R. A function u : C ! R is said to be subharmonic if Z 1 2 u.z/ u.z C eir/d 2 0 for every z 2 C and every r 2 R. Equivalently, u 2 C 2.C/ is subharmonic if and only if u 0. Similarly, the mapping u : Mn.C/ ! R is said to be subharmonic if Z 1 2 u.A/ u.A C eiB/d 2 0 for every A, B 2 Mn.C/. Section 5.4 Subharmonic Functions on Matrix Algebras 169

Fix n 1. For a given subharmonic function u : C ! R, we deﬁne a function uO : Mn.C/ ! R by setting X u.A/O :D u./, A 2 Mn.C/. 2.A/

Here, .A/ means the spectrum of A counted with algebraic multiplicity.

In this section we prove that the function uO : Mn.C/ ! R is subharmonic if u : C ! R is subharmonic. We start with the following lemma which shows that uO is harmonic if u is harmonic.

Lemma 5.4.1. Let A, B 2 Mn.C/ and let D Dfz 2 C : jzj kAk1 CkBk1. For every harmonic function u : D ! R, we have Z 1 2 u.A/O D u.AO C eiB/d. 2 0 Proof. Let f : D ! C be the mapping deﬁned by setting f.z/ :D zn. Writing

Xm i m m ik .A C e B/ D A C e Ck, Ck 2 Mn.C/, kD1 we obtain Xm i m m ik Tr..A C e B/ / D Tr.A / C Cke . kD1

Thus, Z Z 1 2 1 2 f.A/O D Tr.An/ D Tr..A C eiB/n/d D f.AO C eiB/d. 2 0 2 0 By linearity, the equality Z 1 2 f.A/O D f.AO C eiB/d (5.7) 2 0 holds for every polynomial f : D ! C. Recall that for every analytic function f : D ! C, its Taylor polynomials converge to f uniformly on the compact subsets of D. Hence, the equality (5.7) holds for an arbitrary analytic function f : D ! C. Taking real parts, we obtain the equality (5.7) for the real part of an arbitrary analytic function f : D ! C. Since every harmonic function u : D ! C is a real part of an analytic function, the assertion follows. 170 Chapter 5 Commutator Subspace

Informally speaking, the next lemma proves subharmonicity of the function A ! log.jdet.A/j/. We will need the well-known fact that, for A 2 Mn.C/,wehave Y det.A/ D . 2.A/

Lemma 5.4.2. For all matrices A, B 2 Mn.C/, we have Z 1 2 log jdet.A/j log.jdet.A C eiB/j/d. (5.8) 2 0

Proof. The roots of the polynomial ./m 1aree2ki=m,0 k m. Hence, for every 2 C and every m 2 N,wehave ˇ ˇ ˇ mY1 ˇ ˇ 2 ˇ ˇ .1 C e ki=m/ˇ Dj./m 1j. kD0

If 2 C and jj¤1, then the set f1 C ei, 2 .0, 2/g is separated from 0 and, therefore, the mapping ! log.j1 C eij/ is continuous. Hence, for every 2 C with jj¤1, we have

Z 2 2 mX1 log.j1 C eij/d D lim log.j1 C e2ki=mj/ 0 m!1 m kD0 ˇ ˇ ˇ mY1 ˇ1=m ˇ 2 ˇ D 2 lim log ˇ .1 C e ki=m/ˇ m!1 kD0 D 2 log. lim j./m 1j1=m/ D 2 maxflog.jj/,0g. m!1

For every with jjD1, we have D ei0 and, therefore, Z Z Z 2 2 2 log.j1 C eij/d D log.j1 C ei.C0/j/d D log.j1 C eij/d D 0. 0 0 0

The latter equality can be found, for example, in [1, p. 78]. For every C 2 Mn.C/,wehave X X log.jdet.1 C eiC/j/ D log.jj/ D log.j1 C eij/. 2.1CeiC/ 2.C/ Section 5.4 Subharmonic Functions on Matrix Algebras 171

Therefore, Z Z 2 X 2 log.jdet.1 C eiC/j/d D log.j1 C eij/d 0. (5.9) 0 0 2.C/

If det.A/ D 0, then the left-hand side of (5.8) is 1 and the assertion is trivial. If det.A/ ¤ 0, then A is an invertible matrix. Setting C :D A1B,wehaveACeiB D A.1 C eiC/. Therefore,

log.jdet.A C eiB/j/ D log.jdet.A/j/ C log.jdet.1 C eiC/j/.

The estimate (5.8) follows from the above equality and (5.9). The following theorem, providing a representation of a subharmonic function in terms of a harmonic function and a ﬁnite Borel measure on the closure of the domain, is due to Riesz [194, 195]. We refer the reader to the book [247, Section 3.10] for the proof.

Theorem 5.4.3. For every subharmonic function u : C ! R and every domain D C, there exists a ﬁnite countably additive Borel measure on DN (the closure of the domain D in the complex plane) such that Z

u.z/ D u0.z/ C log.jz j/d./, z 2 D, (5.10) DN with u0 being a harmonic function in D. We also need a special form of Fubini’s theorem.

Theorem 5.4.4. Let u be a measurable function on Œ0, 1 Œ0, 1 such that uC 2 L1.Œ0, 1 Œ0, 1/. It follows that Z Z Z Z Z 1 1 1 1 u.s, t/dsdt D u.s, t/dtds D u.s, t/dsdt. 0 0 0 0 Œ0,1Œ0,1 Proof. We have Z Z Z Z Z 1 1 1 1 uC.s, t/dsdt D uC.s, t/dtds D uC.s, t/dsdt < 1 0 0 0 0 Œ0,1Œ0,1 by Fubini’s theorem. We have Z Z Z Z Z 1 1 1 1 u.s, t/dsdt D u.s, t/dtds D u.s, t/dsdt 0 0 0 0 Œ0,1Œ0,1 by Tonelli’s theorem. Subtracting those equalities, we obtain the assertion. 172 Chapter 5 Commutator Subspace

Now, we can prove the main result of this section.

Theorem 5.4.5. For every subharmonic function u : C ! R, the function uO : Mn.C/ ! R is subharmonic. Proof. Using the formula that the determinant of a matrix is equal to the product of its eigenvalues, we obtain ˇ ˇ X X ˇ Y ˇ ˇ ˇ log.jj/ D log.jj/ D log ˇ ˇ D log.jdet.A/j/. 2.A/ 2.A/ 2.A/

Fix a domain D Dfz 2 C : jzj kAk1 CkBk1.UsingTheo- rem 5.4.3, select functions u0 and v such that u D u0 C v, u0 is harmonic in D and v is a potential (the second term in (5.10)). Therefore, Z X Z v.A/O D log.j j/d./ D log.jdet.A /j/d./. DN DN 2.A/ C n For every C 2 Mn. /,wehavejdet.C /jkC k1. For every 2 D, setting C :D A C eiB , we obtain

i i logC.jdet.ACe B /j/ n logC.kACe B k1/ n logC.2kAk1C2kBk1/ where we have set

logC.x/ :D maxflog.jxj/,0g.

Integrating the above inequality, we obtain Z i logC.jdet.A C e B /j/d./d < 1. (5.11) DN Œ0,2 Using (5.11) and Theorem 5.4.4, we obtain Z Z Z 2 2 v.AO C eiB/d D log.jdet.A C eiB /j/d./d 0 0 DN Z Z 2 D log.jdet.A C eiB /j/dd./ DN 0 Z 2 log.jdet.A /j/d./. DN The last inequality uses Lemma 5.4.2. Therefore, Z 1 2 v.AO C eiB/d Ov.A/. (5.12) 2 0 Section 5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace 173

It follows from Lemma 5.4.1 that Z 2 1 i uO0.A C e B/d Ou0.A/. (5.13) 2 0 Adding (5.12) and (5.13) proves the assertion.

5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace

Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. In this section we extend Theorem 1.1.22 by showing that an operator A 2 E is the sum of a normal operator N 2 E with .A/ D .N / and a quasi-nilpotent operator Q 2 E that belongs to the commutator subspace, Q 2 Com.E/. The consequences are that every quasi-nilpotent operator Q 2 E belongs to the commutator subspace, and eigenvalue sequences map the symmetrically normed ideal into its Calkin space, : E ! E.

Theorem 5.5.1. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E and let A 2 E.ThenA D N C Q for a compact normal operator N 2 E with .A/ D .N / 2 E and a quasi-nilpotent operator Q 2 Com.E/.

The proof is given below. We caution the reader that this result is not true for all two-sided ideals of compact operators. In [81], Kalton and Dykema developed the notion of a geometrically stable ideal, a condition which the symmetrically normed ideals satisfy, and gave an example of a non-geometrically stable two-sided ideal of compact operators J and a quasi-nilpotent operator Q 2 J with Q 62 Com.J /. The crucial step in the proof of Theorem 5.5.1 is Theorem 5.5.7 below, which estimates the sum of the eigenvalues of the real (or imaginary) part of a quasi-nilpotent operator Q over a punctured plane in terms of the eigenvalues of 2ejQj . The proof of Theorem 5.5.7 will be based on the properties of subharmonic functions, as in Sec- tion 5.4, and some results on the spectra of matrices, as given in Appendix A.1. The aim of the following lemmas is to split the proof of Theorem 5.5.7 into manageable parts. First we consider nilpotent matrices, and then use the fact that a compact quasi- nilpotent operator can be approximated uniformly by nilpotent matrices. Set

logC.x/ :D maxflog.jxj/,0g, x 2 R.

Lemma 5.5.2. There exists a subharmonic function u 2 C 2.C/ such that

C ju.z/ C

Proof. Deﬁne an increasing function ' 2 C 2.1, 1/ by setting 8 <ˆ0, t<0 '.t/ :D 6t 5 15t 4 C 10t 3,0 t 1 :ˆ 1, t>1.

Deﬁneafunction 2 C 2.1, 1/ by setting 00.t/ D et .2'0.t/ Cj'00.t/j/ and 0.0/ D .0/ D 0. In particular, we have .t/ D 0fort<0and .t/ D .1/ C 0.1/.t 1/ for t>1. Let the function u : C ! R be deﬁned by the formula

u.z/ :D .log.jzj//

Clearly, u 2 C 2.C/. We claim that u 0. Indeed, we have

. .log.jzj/// D . ı log/00.jzj/ Cjzj1. ı log/0.jzj/ Djzj2 00.log.jzj// Djzj1.2'0.log jzj/ Cj'00.log.jzj//j/ and

It follows that u 0 and, therefore, the function u is subharmonic. The estimate (5.14) follows from easy computation.

Theorem 5.4.5 on lifting subharmonic functions to matrices is the central component of proving the next lemma.

Lemma 5.5.3. For every nilpotent operator Q 2 Mn.C/, we have 0 1 Z 2 X 1 @ A c < d 300logC.2ejQj/. 2 0 2.QCeiQ/,jj>1

Proof. Let u be the subharmonic function deﬁned in Lemma 5.5.2. It follows from (5.14) that X c 1Ci i < 100logC.eQ C e Q / Ou.Q C e Q /. 2.QCeiQ/,jj>1 Section 5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace 175

It follows from Corollary 2.3.16 that

1Ci .eQ C e Q / 2e2.Q/.

Therefore,

c 1Ci c 1Ci c logC.eQ C e Q / logC.jeQ C e Q j/ 2logC.2ejQj/.

For the ﬁrst inequality in the latter formula we used Lemma A.1.1 in the appendix. Therefore, 0 1 Z Z 2 X 2 @ A c i < d 400logC.2ejQj/ u.QO C e Q /d. 0 0 2.QCeiQ/,jj>1

By Theorem 5.4.5, the function uO : Mn.C/ ! R is subharmonic. It follows that Z 2 1 i c u.QO C e Q /d Ou.Q/ 100logC.eQ/. 2 0 Therefore, 0 1 Z 2 X @ A c < d 2u.Q/O C 400logC.2ejQj/. (5.15) 0 2.QCeiQ/,jj>1

Since Q is a nilpotent operator, we have .Q/ Df0g. It now follows from (5.14) that X c c u.Q/O < 100logC.eQ/ D100logC.eQ/. (5.16) 2.Q/,jj>1

The assertion follows from (5.15) and (5.16).

Lemma 5.5.3 and results on sums of eigenvalues in the appendix are used to prove the next technical lemma, which links the sums of eigenvalues on a punctured plane of the real part of a nilpotent matrix Q with the absolute value jQj.

Lemma 5.5.4. For every nilpotent operator Q 2 Mn.C/, we have X c 400logC.2ejQj/. jj>1,2.

Proof. The operator Q C eiQ is normal for every 2 Œ0, 2.Wehave

Q C eiQ D .1 C ei/

Applying Lemma A.1.3 to the operators

i i i A1 :D Q C e Q , A2 :D.1 C e /1 2..1Cei /1 2..1ei/=Q/,jj>1

2.nQCeiQ .1/ C n.1Cei/

It follows from Lemma A.1.2 that ˇ ˇ ˇ X 1 C ei X ˇ ˇ ˇ n < .1/ 2 2 Q 2..1Cei/1 2.21 and ˇ ˇ ˇ X 1 ei X ˇ ˇ ˇ n = .1/. 2 2 Q 2..1ei /=Q/,jj>1 2.2=Q/,jj>1

Therefore, ˇ ˇ ˇ X 1 C ei X i.1 ei/ X ˇ ˇ ˇ 2 2 2.QCeiQ/,jj>1 2.21 2.2=Q/,jj>1

12n2jQj.1/ C n2

Integrating over 2 Œ0, 2, we obtain 0 1 ˇ Z ˇ 2 X X ˇ @ A d 0 i 2.QCe Q /,jj>1 2.21 ˇ X ˇ i ˇ 32n2jQj.1/. 2.2=Q/,jj>1

Taking real parts, we obtain 0 1 Z ˇ 2 X X ˇ ˇ 1 @ A 1 ˇ ˇ < d ˇ 16n2jQj.1/. 2 0 2 2.QCeiQ/,jj>1 2.21 (5.17) Section 5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace 177

It follows from Lemma 5.5.3 and (5.17) that X 1 c c 300logC.2ejQj/ C 16n j j.1/ 316logC.2ejQj/. 2 2 Q 2.21

It follows from Lemma A.1.2 that X c c 316logC.2ejQj/ C n2jQj.1/ 400logC.2ejQj/. 2.1

We now want to lift the above technical result from nilpotent matrices to quasi- nilpotent compact operators. Recall that a compact operator is a uniform limit of matrices (with respect to some selected orthonormal basis of the Hilbert space H ). The ﬁrst component of lifting Lemma 5.5.4 is showing that a quasi-nilpotent compact operator is a uniform limit of nilpotent matrices.

Lemma 5.5.5. Every compact quasi-nilpotent operator in L.H / is a uniform limit of nilpotent operators of ﬁnite rank. In selected orthonormal bases, these operators are ﬁnite-dimensional matrices. Proof. Let Q 2 L.H / be a compact quasi-nilpotent operator. One can write Q D U jQj with U being a partial isometry and jQj being a positive compact operator. We have the uniform convergence 1 Q :D U jQjEj j , 1 ! Q. n Q n

Fix >0 and consider the closed set F :Dfz 2 C : jzjg. Clearly, .Q/\F D ;. It follows from [98, 1.2.3 in Chapter 1] that there exists N such that for every n>N we have .Qn/ \ F D;.Since can be taken as arbitrarily small, we obtain

supfjj : 2 .Qn/g!0. (5.18)

? ? Consider the closed subspaces H1n D ker.Qn/ \ Qn.H / and H2n D H1n. By assumption, we have codim.ker.Qn// < 1 and dim.Qn.H // < 1. Hence, codim.H1n/<1 and, therefore, dim.H2n/<1. It is clear that Qn.H / H2n and H1n ker.Qn/. It follows that Qn : H2n ! H2n and Qn : H1n ! 0. By the Schur theorem, every matrix is unitarily equivalent to an upper-triangular matrix. Hence, there exists a unitary operator Un : H2n ! H2n and an upper- 1 triangular operator Tn : H2n ! H2n such that Qn D Un TnUn. The diagonal co- efﬁcients of Tn are precisely the eigenvalues of Qn.LetDn : H2n ! H2n be the diagonal of Tn. It is clear that .Tn Dn/ : H2n ! H2n is a strictly upper-triangular matrix. Hence, .Tn Dn/ : H2n ! H2n is nilpotent and, therefore, so is the matrix

1 1 .Qn Un DnUn/ D Un .Tn Dn/Un. 178 Chapter 5 Commutator Subspace

1 1 We extend the matrix Un DnUn to the operator Un DnUn : H ! H by setting 1 1 Un DnUnx D 0 for every x 2 H1n. It follows from (5.18) that Un DnUn ! 0 1 1 uniformly. Therefore, Qn Un DnUn ! Q uniformly. Since Qn Un DnUn is a ﬁnite-dimensional nilpotent matrix, the assertion follows.

The second component of lifting Lemma 5.5.4 is that, conditionally, sums of eigenvalues of a compact operator over a punctured plane are also approximated by the same sums for matrices.

Lemma 5.5.6. Let A be a compact self-adjoint operator such that 1, 1 … .A/. If An, n 2 N, is a sequence of compact self-adjoint operators such that An ! A uniformly, then X X ! .

2.An/,jj>1 2.A/,jj>1

Proof. Let N1, N2 2 N be such that

.N1, AC/>1 >.N1 C 1, AC/, .N2, A/>1 >.N2 C 1, A/.

It follows that

X XN1 XN2 D .k, AC/ .k, A/. (5.19) 2.A/,jj>1 kD0 kD0

It is clear that .An/C ! AC and .An/ ! A. For all sufﬁciently large n 2 N, we have

k.An/C ACk1 < minf.N1, AC/ 1, 1 .N1 C 1, AC/g and

k.An/ Ak1 < minf.N2, A/ 1, 1 .N2 C 1, A/g.

For every such n, it follows from Lemma 2.3.17 that

.N1, .An/C/>1 >.N1 C 1, .An/C/, .N2, .An//>1 >.N2 C 1, .An//.

It follows that

X XN1 XN2 D .k, .An/C/ .k, .An//. (5.20) 2.An/,jj>1 kD0 kD0 Section 5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace 179

By Lemma 2.3.17, we have that

XN1 XN1 XN2 XN2 .k, .An/C/ ! .k, AC/, .k, .An// ! .k, A/. (5.21) kD0 kD0 kD0 kD0 The assertion follows from (5.19), (5.20) and (5.21).

We are now in a position to lift Lemma 5.5.4 to a statement about compact quasi- nilpotent operators and obtain our main technical result.

Theorem 5.5.7. For every compact quasi-nilpotent operator Q 2 L.H /, we have the estimate ˇ ˇ ˇ X ˇ X ˇ ˇ 400 log./. jj>1,2.1,2.2ejQj/

Proof. By Lemma 5.5.5, there exists a sequence of ﬁnite-dimensional nilpotent ma- N trices Qn, n 2 , such that Qn ! Q uniformly. Hence, Qn ! Q and, therefore, 2 2 1,2.

In general, since 0. It follows that X X c 1 D .1 C / .1 C /logC..1 C / 2ejQj/. jj>1,2.1,2..1C/1

Clearly, as # 0then

c 1 c logC..1 C / 2ejQj/ ! logC.2ejQj/.

Thus, (5.22) holds. Substituting Q !Q, we obtain the assertion of the theorem.

Corollary 5.5.8. For every quasi-nilpotent compact operator Q 2 L.H /, we have the estimate ˇ ˇ ˇ X2n ˇ Yn 1=.nC1/ 1 ˇ ˇ ˇ .k,

Proof. It is clear that, for every n 0, j.2n,

In particular,

2n ..n,2ejQj/gf.k,

Therefore, ˇ ˇ ˇ ˇ ˇ X2n ˇ ˇ X ˇ ˇ ˇ ˇ ˇ ˇ .k, .n,2ejQj/

Applying Theorem 5.5.7 to the quasi-nilpotent operator ..n,2ejQj//1Q, we obtain

ˇ ˇ n ˇ X ˇ X .k,2ejQj/ ˇ ˇ 400.n,2ejQj/ log .n,2ejQj/ 2..n,2ejQj/ kD0 Q . n .k,2ejQj//1=.nC1/ D 400.n C 1/.n,2ejQj/ log kD0 . .n,2ejQj/

Recall that the inequality log.˛/ ˛ holds for every positive ˛. It follows that

ˇ ˇ n ˇ X ˇ Y 1=n ˇ ˇ 400.n C 1/ .k,2ejQj/ . 2..n,2ejQj/ kD0

Hence, X2n Yn 1=.nC1/ j .k,

The assertion follows.

If Q belongs to a two-sided ideal of compact operators J with Calkin sequence space J and Yn 1=.nC1/ 1 .k, Q/ 2 J D kD0 n 0 then Corollary 5.5.8 implies that C.

Lemma 5.5.9. Let E be a symmetric sequence space and let x 2 E. It follows that the sequence Yn 1=.nC1/ 1 .k, x/ 2 E. D kD0 n 0 Proof. For brevity set x D .x/. It follows that, for n 0,

n n n 2Y1 2 nY1 2 k n 2 xk x2 1 x2k1 kD0 kD0 n nY1 2 1n n k 1nC2 2 n k 2 1n k D 4 4 x2 1 .4 x2k1/ 4 max 4 x2k1. 0kn kD0 Set X1 k z :D 4 2k x. kD0 Clearly, z 2 E.Wehave Xn Xn n k kn n k z2 1 4 x2nk1 D 4 x2k1 4 max 4 x2k1. 0kn kD0 kD0

If 2n 1 m 2nC1 1, it follows that

n n Ym 1=.mC1/ 2Y1 2 xk xk 16z2nC11 16zm. kD0 kD0 The assertion follows. We can now prove Theorem 5.5.1. Proof of Theorem 5.5.1. Let A belong to a symmetrically normed ideal of compact operators E and let E be the Calkin sequence space of E. From Theorem 1.1.22 we have that A D N C Q with N being a compact normal operator, Q being a compact quasi-nilpotent operator, and .A/ D .N /. Note also that j.A/jDj.N /jD.N / as N is normal. It follows from Weyl’s lemma (Lemma 1.1.20 in Chapter 1) that

n n n Y 1=.nC1/ Y 1=.nC1/ Y 1=.nC1/ .n, N/ .k, N/ D j.k, A/j .k, A/ . kD0 kD0 kD0 By Lemma 5.5.9, we have .N / 2 E. Hence, N 2 E and .A/ 2 E.SinceQ D A N , therefore, Q 2 E as well. Using Lemma 5.5.9 and the statement of Corollary 5.5.8, 182 Chapter 5 Commutator Subspace we have that C..

5.6 Description of the Commutator Subspace

The results in Sections 5.2–5.5 provide the following extension and noncommutative generalization of the Figiel–Kalton theorem to (continuous) symmetric functionals. It is the main result of this chapter.

Theorem 5.6.1. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. If A 2 E, then (a) A 2 Com.E/ if and only if C..A// 2 E.

(b) A 2 Com.E/ if and only if 1 1 1 C..A// ! 0 log.m/ m m

in the norm topology of E. Here, .A/ denotes an eigenvalue sequence of A.

Proof. By Theorem 5.5.1, A D N C Q with N 2 E a normal compact operator such that .A/ D .N / 2 E,andQ is a quasi-nilpotent operator such that Q 2 Com.E/. Hence, A 2 Com.E/ if and only if N 2 Com.E/. The ﬁrst assertion follows from Theorem 5.2.1 since N 2 Com.E/ if and only if C..N// 2 E. The second assertion follows from Theorem 5.3.2 since N 2 Com.E/ if and only if 1 1 1 C..N// ! 0. log.m/ m m

The ﬁrst corollary of Theorem 5.6.1 is that the statement of the theorem can be extended to differences of operators and differences of eigenvalue sequences. The proof requires the following extension of Lemma 5.2.8 to arbitrary operators in E.

Lemma 5.6.2. If A1, A2 2 E, then

C.A1 C A2/ 2 C.A1/ C C.A2/ C E.

Proof.LetA D N C Q with N , Q as in Theorem 5.5.1. By Lemma 5.2.8, we have

C..

Hence,

C..A// D C..N// 2 C..

For operators A1, A2 2 E,wehave

C..A1 C A2// 2 C..<.A1 C A2/// C iC..=.A1 C A2/// C E

C..

C..A1// C C..A2// C E.

Corollary 5.6.3. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. If A1, A2 2 E, then

(a) A1 A2 2 Com.E/ if and only if C..A1/ .A2// 2 E.

(b) A1 A2 2 Com.E/ if and only if 1 1 1 C..A / .A // ! 0 log.m/ m m 1 2

in the norm topology of E.

Proof. It follows from Theorem 5.6.1 that A1 A2 2 Com.E/ if and only if C..A1 A2// 2 E. By Lemma 5.6.2, C..A1 A2// 2 E if and only if C..A1/ .A2/// 2 E. Similarly for the second assertion, we have A1 A2 2 Com.E/ if and only if 1 1 log.m/ .1 m m/C..A1 A2/ ! 0. However, there is some y 2 E with C..A1 A2// C..A1/ .A2/// D y by Lemma 5.6.2. Then 1 1 2 1 m y kykE ! 0. log.m/ m E log.m/

1 1 1 1 So log.m/ .1 m m/C..A1 A2/ ! 0 if and only if log.m/ .1 m m/C..A1/ .A2// ! 0.

The Lidskii formula for all traces on a symmetrically normed ideal E of the algebra L.H / is now a trivial corollary. Recall that any unitarily invariant linear functional ' : E ! C is called a trace, and called a continuous trace if ' 2 E . By Theorem 5.5.1, .A/ 2 E for any A 2 E, so eigenvalue sequences give a well deﬁned map (for some speciﬁed ordering) : E ! E 184 Chapter 5 Commutator Subspace between a symmetrically normed ideal of compact operators and its Calkin sequence space. Also recall from Theorem 4.6.2 that, unless E is the ideal of trace class operators, all continuous traces on E are singular. The Lidskii formula is the statement that a trace on E is completely determined by eigenvalues (alternatively, it is completely determined by its restriction to the diagonal of the ideal E).

Corollary 5.6.4. Let E be a symmetrically normed ideal of the algebra L.H / and let E be the Calkin sequence space of E. (a) If ' is a trace on E,then

'.A/ D ' ı diag..A//, A 2 E.

(b) If ' is a continuous trace on E, there exists a unique symmetric functional f on E such that ' D f ı

and f D ' ı diag.

Here denotes eigenvalue sequences and diag denotes the diagonal operator with respect to any orthonormal basis of H .

Proof. By the previous corollary A diag..A// 2 Com.E/ since A and diag..A// can have the same eigenvalue sequence (namely, .A/). The ﬁrst assertion follows since Com.E/ is the common kernel of all traces. The second assertion follows from the ﬁrst assertion and Theorem 4.4.1.

Another corollary of Theorem 5.6.1 is simple criteria for the existence of traces on E in terms of invariance of the Calkin sequence space E under the Cesàro operator C : l1 ! l1. Criteria for existence of continuous traces in terms of the Calkin sequence space were already noted in Section 4.6 in the preceding chapter.

Corollary 5.6.5. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E.ThenE admits a nontrivial trace if and only if C.E/ 6 E.

Proof. Supposethat there exists some 0

Example 5.6.6. Let 1

Proof. It is easily checked that if C : l1 ! l1 denotes the Cesàro operator, then C.lp/ lp and C.lp,1/ lp,1.

Example 5.6.7 (Lidskii’s Formula). Let L1 be the ideal of trace class operators and Tr the canonical operator trace introduced in Section 1.3.1 of Chapter 1.Then X1 Tr.A/ D .n, A/, A 2 L1, nD0 where .A/ 2 l1 is any eigenvalue sequence of A.

Proof. From Corollary 5.6.4 we have that Tr.A/ D Tr ı diag..A//. Choosing an 1 orthonormal basis fengnD0 of H , X1 X1 Tr ı diag..A// D hen, emi.m, A/hem, eniD .n, A/. n,mD0 nD0

The canonical operator trace Tr is not the only trace on the trace class operators L1.

Example 5.6.8. The canonical trace Tr is, up to scalar constant, the unique continuous trace on L1. However, there exists a trace ' : L1 ! C (which is not continuous) such that ' 6D Tr.

Proof. Theorem 4.10.1 indicates that L1 can admit only the continuous trace Tr up to a constant. Hence, ker Tr D Com.L1/. If A 2 L1 is a diagonal operator such that Tr.A/ D 0, then there exist ﬁnite rank diagonal operators An ! A (in L1), n 0, such that Tr.An/ D 0. In particular, C.An/ is a sequence with ﬁnite support. Hence, An 2 Com.L1/ and, therefore, A 2 Com.L1/. On the other hand, take a 2 l1 such that 1 a.n/ D , n 1, n log2.1 C n/ P and a.0/ D k1 a.k/. It follows that 1 .C a/.n/ , n 1, n log.1 C n/ and, therefore, Ca … l1.IfA D diag.a/,thena D .A/ and, therefore, C.A/ … l1. This implies that A … Com.L1/. So Com.L1/ is strictly contained in ker Tr. The assertion follows. 186 Chapter 5 Commutator Subspace

The next example demonstrates that it is possible for Com.E/ to be a dense strict subspace of a symmetrically normed ideal E, so that the ideal E admits nontrivial traces but does not admit a nontrivial continuous trace. The example involves an Orlicz ideal of compact operators. Let M.t/ be a convex function on Œ0, 1/ such that M.t/ > 0forallt>0andsuch that M.t/ t 0 D M.0/ D lim D lim . t!0 t t!1 M.t/

Denote by lM the Orlicz sequence space, see e.g. [139, 150],

lM :Dfx 2 l1 : kxklM < 1g where kxklM is deﬁned by

n X1 o 1 kxklM :D inf s>0: M.jxnj=s/ 1 , x DfxngnD0 2 l1. nD0

Equipped with the Fatou norm kxklM , lM is a fully symmetric sequence space.

Example 5.6.9. There exists a symmetrically normed ideal of compact operators L . ˆ, kkLˆ /, corresponding to an Orlicz space lˆ, L L ˆ :DfA 2 .H / : .A/ 2 lˆg , kAkLˆ :Dk.A/klˆ such that Com.Lˆ/ is a strict subspace of Lˆ and Com.Lˆ/ D Lˆ.

Proof. There exists an Orlicz sequence space lˆ which is not an interpolation space between lp and l1 for any p>1 [149]. For such a space, we have C : lˆ 6! lˆ, and hence Com.Lˆ/ is a strict subset of Lˆ. However, for any Orlicz space lˆ 1 with lˆ ¤ l1,wehavethat n nx ! 0asn !1for every x 2 lˆ [240]. By Theorem 4.9.2, Lˆ has only trivial positive symmetric functionals, and due to the Jordan decomposition of symmetrical functionals, we obtain that Lˆ admits no nontrivial symmetric functionals and hence no continuous traces. From Lemma 5.3.3, Com.Lˆ/ D Lˆ. In the next chapter we (re-)introduce Dixmier traces as continuous singular traces on Lorentz ideals. Part III examines the Lidskii formula, and variants of it, for Dixmier traces. Section 5.7 Commutator Subspace of the Weak Ideal L1,1 187

5.7 Commutator Subspace of the Weak Ideal L1,1

Let L1,1 be the two-sided ideal of the algebra L.H / corresponding to the weak-l1 sequence space l1,1,

L1,1 :DfA 2 L.H / : .A/ 2 l1,1g, as described in Example 1.2.9 in Chapter 1. It is not a symmetrically normed ideal of compact operators (it is quasi-normed, but not normed), but one can deﬁne and describe its commutator subspace

Com.L1,1/ :D LinfŒA, B : A 2 L1,1, B 2 L.H /g.

Theorems 5.7.4 and 5.7.6, which are the analogs of Theorems 5.5.1 and 5.6.1 respectively, are the main result in this section. We use them in the applications in Part IV. The quasi-norm on L1,1 is given by

kAk1,w :D sup.n C 1/.n, A/. n0

In what follows, we need the mapping deﬁned by setting

Xn : A ! .k, A/ C l1, A 2 L1,1. (5.23) kD0 n 0

The description of the commutator subspace of L1,1 relies on showing the linearity of the mapping . We ﬁrst show the linearity of on the self-adjoint operators in L1,1, which will be sufﬁcient to repeat Theorem 5.2.1.

Lemma 5.7.1. The mapping in (5.23) is linear on the self-adjoint part of L1,1.

Proof. If A1, A2 2 L1,1 are positive operators, then

Xn Xn .k, A1 C A2/ D .k, A1 C A2/ kD0 kD0 Xn Xn ..k, A1/ C .k, A2// D ..k, A1/ C .k, A2// kD0 kD0 188 Chapter 5 Commutator Subspace and

Xn Xn 2XnC1 ..k, A1/ C .k, A2// D ..k, A1/ C .k, A2// .k, A1 C A2/ kD0 kD0 kD0 C Xn 2Xn 1 1 .k, A C A / CkA C A k 1 2 1 2 1,w k C 1 kD0 kDnC1 Xn .k, A1 C A2/ C log.2/ kA1 C A2k1,w. kD0

Hence, the mapping (5.23) is additive on the positive cone of L1,1. Hence, the mapping Xn A ! ..k, AC/ .k, A// C l1 (5.24) kD0 n 0 is linear on the self-adjoint part of L1,1. It follows from Lemma 5.2.7 that

Xn Xn ..k, AC/ .k, A// D .k, A/ C O.1/ kD0 kD0 for every self-adjoint A 2 L1,1. Hence, the linear mapping in (5.24) coincides with and the assertion follows.

Having shown that is linear on self-adjoint operators, the proof of the following result is identical to that of Theorem 5.2.1 and is, therefore, omitted.

Lemma 5.7.2. Let A 2 L1,1 be a normal operator. Then A 2 Com.L1,1/ if and only if .A/ D l1.

To extend the statement of Lemma 5.7.2 to all operators in L1,1 we must consider the decomposition of an operator into normal and quasi-nilpotent parts as before.

Lemma 5.7.3. Let Q 2 L1,1 be quasi-nilpotent. Then .

Proof. By Lemma 5.5.8 ˇ ˇ ˇ X2n ˇ Yn 1=.nC1/ 1 ˇ ˇ ˇ .k,

Since .k,2ejQj/ 2ekQk1,w=.k C 1/ for k 0, we obtain that ˇ ˇ ˇ X2n ˇ 1 ˇ ˇ 1=.nC1/ 2 1 ˇ .k,

Theorem 5.7.4. If A 2 L1,1, then A D N C Q where N 2 L1,1 is normal with .A/ D .N / 2 l1,1 and Q 2 Com.L1,1/ is quasi-nilpotent. Proof. Let A D N C Q where N , Q are as in Theorem 1.1.22. It follows from Lemma 1.1.20 that Yn 1=.nC1/ Yn 1=.nC1/ j.n, A/j j.k, A/j .k, A/ . kD0 kD0

Since .k, A/ kAk1,w=.k C 1/ for k 0, we obtain 1 j.n, A/jkAk ..n C 1/Š/1=.nC1/ ekAk . 1,w 1,w n C 1

Here, the last inequality follows from Stirling’s formula. Hence, .A/ 2 l1,1.Since j.A/jD.N / then N 2 L1,1. It follows also that Q 2 L1,1. We now show that Q 2 Com.L1,1/.LetQ D

A corollary of this result is that is linear on L1,1.

Lemma 5.7.5. The mapping in (5.23) is linear on L1,1.

Proof. We ﬁrst show that is linear with respect to the decomposition A D

So .A/ D .N/ D .

Now, if A1, A2 2 L1,1,

.A1 CA2/ D .<.A1 CA2//Ci.=.A1 CA2// D .

It follows from Lemma 5.7.1 that

Therefore,

.A1 C A2/ D .

As in Section 5.6, we use the term trace to mean a unitarily invariant linear functional on L1,1.SinceL1,1 is not a symmetrically normed ideal we are not providing statements about continuous traces.

Theorem 5.7.6. Let L1,1 be the two-sided ideal of the algebra L.H / corresponding to the sequence space l1,1. P L L n (a) If A 2 1,1,thenA 2 Com. 1,1/ if and only if kD0 .k, A/ D O.1/. P L L n (b) IfPA1, A2 2 1,1,thenA1 A2 2 Com. 1,1/ if and only if kD0 .k, A1/ n kD0 .k, A2/ D O.1/.

(c) If ' : L1,1 ! C is a trace, then '.A/ D ' ı diag..A// for every A 2 L1,1. Here denotes eigenvalue sequences and diag denotes the diagonal operator with respect to any orthonormal basis of H .

Proof. Observe that C..A// 2 l1,1 and .A/ D l1 are equivalent conditions.

(a) Let A D N C Q with N 2 L1,1 a compact normal operator, .A/ D .N /, and Q 2 Com.L1,1/ a compact quasi-nilpotent operator, as in Theorem 5.7.4. Then we have that A 2 Com.L1,1/ if and only if N 2 Com.L1,1/.Bydesign .A/ D .N/,so.A/ D 0 if and only if .N/ D 0. The result follows from Lemma 5.7.2.

(b) By part (a), A1A2 2 Com.L1,1/ if and only if .A1 A2/ D 0. By Lemma 5.7.5 this occurs if and only if .A1/ D .A2/.

There are nontrivial traces on L1,1 since C.l1,1/ 6 l1,1 (although we knew already that nontrivial traces existed since Section 1.3.2 established the existence of Dixmier traces on L1,1 M1,1). Section 5.7 Commutator Subspace of the Weak Ideal L1,1 191

The criteria .A/ D 0, C.A/2 l1,1,and

Xn .j , A/ D O.1/, n 0, j D0 are equivalent. Therefore, from this condition, we have that C.l1/ l1,1 and, hence, from Theorem 5.7.6 (a), we obtain that

L1 Com.L1,1/.

The following corollary is immediate if we deﬁne a trace ' : L1,1 ! C to be singular if it vanishes on the ﬁnite rank projections (or the trace class operators).

Corollary 5.7.7. Every trace on L1,1 is singular.

In fact, L1,1 is the smallest two-sided ideal of compact operators such that all (nontrivial) traces are singular.

Theorem 5.7.8. Let E be a symmetrically normed ideal of compact operators. Then the following statements are equivalent. (a) Every trace ' : E ! C is singular.

(b) The ﬁnite rank operators belong to the commutator subspaceof E, C00 Com.E/.

(d) L1,1 E. Proof. Let E be the Calkin sequence space of E. (a) ) (b). By assumption, every trace ' : E ! C is singular, that is, it vanishes on C00. Hence, C00 Com.E/ since the commutator subspace is the common kernel of all traces. C E 1 1 (b) ) (d). If 00 Com. / then Cc00 E. Hence, f.1Cn/ gnD0 DC.1, 0, :::/2 E. It follows that l1,1 E. Hence L1,1 Com.E/. (d) ) (c). If L1,1 E,thenL1 Com.L1,1/ Com.E/. (c) ) (a). Since C00 L1 Com.E/, then all traces on E are singular. For a symmetrically normed ideal E of compact operators, compare the statement above that, all traces on E are singular traces, that is,

C00 Com.E/,

if and only if L1,1 E, with the statement for continuous traces shown in Chapter 4, 192 Chapter 5 Commutator Subspace

all continuous traces on E are singular continuous traces, that is,

E0 Com.E/

if and only if L1 6D E.

5.8 Notes

Commutator Subspaces Paul Halmos in 1952 and 1954 [106,107] observed that every bounded operator on an infinite- dimensional Hilbert space is the sum of two commutators, and it is also likely that he made the first use of a commutator subspace to show a result for traces, i.e. the consequence of the observation is that L.H / admits no nontrivial traces when H is infinite-dimensional. In 1971 Carl Pearcy and David Topping [179] initiated the study of commutator subspaces of Schatten–von Neumann ideals of compact operators. We refer to the survey article of Gary Weiss [254] for additional background. Halmos [107, p. 198] questioned what the spectrum of commutators look like. Our modern statement, involving eigenvalues and the Cesàro operator, originated with Nigel Kalton in 1989 [124]. Precursors to this result can be seen in Weiss’ papers [252] and [253]. Kalton showed Theorem 5.1.3(a) for the trace class operators L1. Incidentally, this showed that the commutator subspace Com.L1/ was smaller than the kernel of the canonical trace Tr. Hence, there exist traces (but not continuous traces) other than Tr on L1 (Example 5.6.8). This outcome was Kalton’s motivation for studying Com.L1/. Section 5.2 follows Dykema, Figiel, Weiss and Wodzicki [80]. In [80] the statement of Theorem 5.2.1 (Theorem 5.1.3 (a) for normal operators) was proved for every two-sided ideal of compact operators. The result of [80] combined with Kalton’s observation in [125] proves Theorem 5.1.3 (a) without any context to symmetrically normed ideals except invok- ing Lemma 5.5.9. Many other corollaries of commutator results, and answers to questions of Halmos, Pearcy and Topping, are shown in [80]. The way we have stated Theorem 5.1.3 appeared first in [125]. Restricting to symmetrically normed ideals enables proofs for normal operators to take a more direct route than the general case in [80]. Section 5.3 follows the results in Chapter 4.

Commutator Subspace of the Weak-l1 Ideal We showed Lemma 5.7.2 directly for the non-symmetrically normed ideal L1,1. Showing it just for this ideal of compactoperators is substantially easier than the general result of Dykema, Figiel, Weiss and Wodzicki [80, Theorem 5.1]. The ideal L1,1 plays a special role in commutator subspace theory as explained in [80]. See also [121]. Besides Theorem 5.7.8 we have not elaborated on the role that L1,1 plays, our interests lie henceforth in Dixmier traces rather than the structure of ideals of compact operators. See [80, Section 5] for other examples of ideals that do and do not support nontrivial traces. Section 5.8 Notes 193

Lidskii Formula Background to Lidskii’s original formula has already been mentioned in Chapter 1. Lidskii in 1959 [148] proved that the trace of every trace class operator is given by the sum of its eigenvalues. In 1998, Nigel Kalton [125], and Kalton and Ken Dykema [81] used the commutator subspace result of Dykema, Figiel, Weiss and Wodzicki (which appeared chronologically later [80]) to prove the Lidskii formula for every trace on every geometrically stable two-sided ideal J of compactQ operators. If J is the Calkin sequence space of J , geometric stability is n 1=.nC1/ the condition that f. kD0 xk/ g2J for every x 2 J (c.f. Lemma 5.5.9). Sections 5.4 and 5.5 follow Kalton and Dykema. Theorems 5.7.4 and 5.7.6 are true for all geometrically stable ideals. The way we have stated the Lidskii theorem appeared ﬁrst in [126]. An extension of the Lidskii theorem for non-geometrically stable ideals appears in [126, Corollary 4.7]. With regard to Dixmier traces, we return to the Lidskii formula in Chapter 7. In that chapter we prove variants of the formula relating to heat kernel and -function residue formulations of Dixmier traces. Chapter 6 Dixmier Traces

6.1 Introduction

So far we have discussed the existence of continuous traces on symmetric operator spaces of general semifinite (atomic or atomless) von Neumann algebras, and the existence of traces, continuous or otherwise, on the symmetric operators spaces of the atomic von Neumann algebra L.H / (which are, except for L.H / itself, the symmetrically normed ideals of compact operators). Here, H is a separable Hilbert space. In this chapter we consider the construction of traces. In this chapter, as in the last, we consider only the atomic factor M D L.H / of the set of all bounded operators on H . We indicate when the same statements are true for the general semifinite (atomic or atomless) case. We have used the Calkin correspondence between a symmetric ideal E of the algebra L.H / and its corresponding symmetric sequence space E to lift the set of symmetric functionals on E to obtain the set of all continuous traces on E. Specifically, every continuous trace ' 2 E is of the form

' D f ı .A/, A 2 E for a symmetric functional f 2 E,where.A/ denotes an eigenvalue sequence of the compact operator A. However, the Lidskii formula in the above display does not provide, in itself, concrete formulas for continuous traces. We know explicitly how to construct the trace on the ideal of trace class operators, E D L1. Summation is the only symmetric functional on l1 (up to a constant) and yields the only (up to a constant) continuous trace Tr on L1. The question remains how to construct traces on other symmetric ideals. At this level of generality the question is open; remarkably, however, the construction of all fully symmetric traces on a fully symmetric ideal is known. Recall that a positive trace ' 2 E is fully symmetric if it respects Hardy–Littlewood submajorization, that is, if 0 A, B 2 E and A B then '.A/ '.B/. We say remarkably because the first construction of singular traces, Jacques Dixmier’s 1966 construction [62], which we introduced in Section 1.3.2 of Chapter 1, turns out to provide the general construction of all fully symmetric traces on a fully symmetric ideal of compact operators excepting the trace class operators. In this chapter we look initially at the Lorentz ideals of compact operators introduced in Example 1.2.7 in Chapter 1. If : RC ! RC is an increasing positive Section 6.1 Introduction 195 concave function then Xn M L 1 :DfA 2 .H / : kAkM :D sup .k, A/ < 1g n0 .n C 1/ kD0 defines a fully symmetric ideal of compact operators with a Fatou norm. We show that, if .2t/ lim inf D 1 (6.1) t!1 .t/ is satisfied, then the Lorentz ideal M possess an infinite number of continuous traces, otherwise none. In particular, such a Lorentz ideal of compact operators always admits a singular continuous trace Tr!, called a Dixmier trace, defined by setting 1 Xn Tr .A/ :D ! .k, A/ ,0 A 2 M . (6.2) ! .n C 1/ kD0

Here, ! is some dilation invariant extended limit on the algebra l1 of all bounded sequences. This, essentially, was Dixmier’s original result, with the extension completely to the condition (6.1) performed in the recent paper [127]. In [127], the following unexpected result was established, showing that the set of normalized fully symmetric traces on a Lorentz ideal of compact operators and the set of Dixmier traces on that Lorentz ideal are identical. The fully symmetric functionals are therefore explicitly constructible, which is the ﬁrst main result of this chapter.

Theorem 6.1.1. If satisfies the condition (6.1), then every normalized fully symmetric trace on M is a Dixmier trace Tr! for some !. Comments we make in this chapter, and a discussion in the end notes, indicate how to adjust the definition of a Dixmier trace so that a similar result holds for a Lorentz operator space associated to a semifinite atomless von Neumann algebra. In [259], Mariusz Wodzicki considered multiplicative renormalization of positive compact operators as part of a general construction of traces on symmetric ideals of compact operators different from Lorentz ideals. More precisely, given a positive function : RC ! RC (Wodzicki did not assume this function to be either increasing or concave), one can construct a mapping 1 Xn A ! .k, A/ . .n C 1/ kD0 n 0

Applying some limiting procedure to the latter sequence (Wodzicki called this “multiplicative renormalization” and used Stone–Cechˇ compactiﬁcation for this purpose), 196 Chapter 6 Dixmier Traces we are left with the question as to whether this construction produces a trace. If it does, then it is natural to refer to such a trace also as a Dixmier trace. Wodzicki proved a criterion for the additivity of multiplicative renormalization (see Theorem 3.4 in [259]). The questions of ﬁniteness and nontriviality of multiplicative renormalization, also considered in [259], happen to be harder. It is proved in [259] that, for every principal ideal, multiplicative renormalization produces a trace if and only if the ideal admits a trace. The latter result still relates to the realm of Lorentz ideals and can be compared with [67, 103, 127,245]. The main result of this chapter extends the above mentioned results of Wodzicki and Theorem 6.1.1 to an arbitrary fully symmetric ideal of L.H /.

Theorem 6.1.2. Let E ¤ L1 be a fully symmetric ideal of L.H /.IfE admits a continuous trace, then there are Dixmier traces on E. Moreover, the set of those Dixmier traces is weak dense in the set of all fully symmetric traces on E.

6.2 Extended Limits

Extended limits are extensions to l1 of the usual limit functional lim acting on the space c of all convergent sequences. The limit functional lim : c ! C is deﬁned by setting

lim.x/ :D lim x.n/, x Dfx.n/gn0 2 c, n!1 and it is a continuous linear functional on c. By the Hahn–Banach theorem, the functional lim admits at least one linear extension Lim : l1 ! C such that jLim.x/j kxk1 for all x 2 l1. It turns out that there are many extensions of lim to l1.Wecall such an extension an extended limit.

Deﬁnition 6.2.1. A positive linear functional ' on a von Neumann algebra M is called a state if '.1/ D 1.

Remark 6.2.2. Extended limits are states on l1,i.e.Lim.1/ D 1, and are characterized by their vanishing on c0.

Proof. Suppose ' is a state on l1 which vanishes on c0.Ifx 2 c,thenx lim.x/1 2 c0. Hence, '.x/ D lim.x/ '.1/ D lim.x/. The converse, that an extension Lim is a state and vanishes on c0, is obvious and omitted.

Every state on l1 is continuous, so it is sufficient for a state to be an extended limit if it vanishes on every finitely supported sequence x 2 l1. This will be our definition of an extended limit. We define also the parallel notion of an extended limit on L1.0, 1/. Section 6.2 Extended Limits 197

Definition 6.2.3. Let ! be a state on the von Neumann algebra l1 (respectively, L1.0, 1/). Then ! is called an extended limit if it vanishes on every finitely supported x 2 l1 (respectively, finitely supported x 2 L1.0, 1/). Various kinds of extended limits will be relevant to the development of the theory of Dixmier traces and its applications. We introduce dilation invariant extended limits here, and will consider other types of invariant extended limits in Part III. The dilation semigroup s : L1.0, 1/ ! L1.0, 1/, s>0, acts by the formula

.sx/.t/ :D x.t=s/, t>0, x 2 L1.0, 1/.

Similarly, the discrete dilation semigroup n : l1 ! l1, n 2 N, acts by the formula

n.x0, x1, :::/:D .x„0, :::ƒ‚, x…0, x„1, :::ƒ‚, x…1, /, x 2 l1. n times n times

Deﬁnition 6.2.4. A state ! on the von Neumann algebra l1 (respectively, L1.0, 1/) is called dilation invariant if ! ı n D !, n 2 N, (respectively, ! ı s D !, s>0). To prove that dilation invariant extended limits exist, we use the following invariant version of the Hahn–Banach theorem. Its proof can be found in standard analysis texts, e.g. [83, Theorem 3.3.1]. Functions and sequences in the following results are real- valued.

Theorem 6.2.5. Let X be a linear space. Given (a) an action g : x ! g.x/, g 2 G, x 2 X, of a commutative semigroup G on X (b) a G-invariant subspace Y of X (c) a convex homogeneous functional p : X ! R such that p ı g p for every g 2 G (d) a G-invariant linear functional ! : Y ! R such that ! p then there exists a G-invariant extension ! : X ! R such that ! p.

Corollary 6.2.6. Dilation invariant extended limits exist on l1 and on L1.0, 1/.

Proof. The dilation semigroup n, n 2 N, is an action of the commutative multiplicative semigroup N on the space l1. The subspace c is n-invariant for every n 2 N. The convex functional p : x ! lim sup.x/ is also dilation invariant.

Deﬁne a linear functional ! on c by setting !.1/ D 1and!jc0 D 0. So deﬁned ! is dilation invariant and, by Theorem 6.2.5 admits a dilation invariant extension on l1 such that !.x/ lim sup.x/, x 2 l1.Wehavethat,

lim inf.x/ Dlim sup.x/ !.x/ lim sup.x/, x 2 l1.

In particular, ! is a state on l1 that vanishes on c0. The assertion is proved. The proof for L1.0, 1/ is identical, hence it is omitted. 198 Chapter 6 Dixmier Traces

In the following sections, we will need a more explicit way of constructing dilation invariant extended limits on L1.0, 1/. The following lemma is frequently used in this and the next chapter. The functional on L1.0, 1/, deﬁned by the formula Z 1 NT x.s/ds .x/ :D lim sup sup , x 2 L1.0, 1/, (6.3) T !1 N>0 log.T / N s is convex and homogeneous.

Lemma 6.2.7. If : L1.0, 1/ ! R is given by the formula (6.3) and if ! 2 L1.0, 1/ is such that ! , then ! is a dilation invariant extended limit.

Proof. By the conditions on !,wehavethat

.x/ !.x/ .x/, x 2 L1.0, 1/. (6.4)

Note that .x/ 0 for every 0 x 2 L1.0, 1/. It follows that ! is positive. Noting that !.1/ D 1, we conclude that ! is a state. If x 2 L1.0, 1/ is ﬁnitely supported, then .x/ D .x/ D 0. It follows from (6.4) that !.x/ D 0. Hence, ! is an extended limit. Further, for every x 2 L1.0, 1/ and u>0wehave ˇ Z ˇ ˇ Z Z ˇ ˇ NT ˇ ˇ NT N ˇ ˇ .x ux/.s/ds ˇ ˇ x.s/ds x.s/dsˇ ˇ ˇ D ˇ ˇ. N s NT=u s N=u s

Therefore, 2j log.u/j j.x ux/jlim sup sup kxk1 D 0. T !1 N>0 log.T /

Hence,

!.x ux/ .x ux/ D 0, !.x/ .ux x/ D 0.

Thus, ! is dilation invariant.

6.3 Dixmier Traces on Lorentz Ideals

We show that Dixmier traces exist on a Lorentz ideal M of compact operators if and only if satisﬁes the condition

.2t/ lim inf D 1. (6.5) t!1 .t/ Section 6.3 Dixmier Traces on Lorentz Ideals 199

We begin by recalling the deﬁnition of the Lorentz ideals M from Section 2.6. Deﬁne the Lorentz function space Z 1 t M :D x 2 S.0, 1/ :sup .s, x/ds < 1 , t0 .t/ 0 and the corresponding (fully symmetric) Lorentz ideal of compact operators

M :DfA 2 L.H / : .A/ 2 M gDfA 2 L.H / : .A/ 2 m g.

Observe that all the functions .A/ are bounded when A 2 L.H /. Hence, without loss, we may take such that .t/ D O.t/ as t # 0. This has several implications, the proofs of which are a simple exercise, hence omitted.

Remark 6.3.1. The following statements are equivalent for the concave increasing function : RC ! RC. (a) We have .t/ D O.t/ as t # 0. 0 (b) We have 2 L1.0, 1/.

Deﬁnition 6.3.2. Let M be a Lorentz ideal of the algebra L.H /. Given a dilation invariant extended limit ! on L1.0, 1/, we deﬁne a (a priori, nonlinear) functional on the positive cone of M by setting Z 1 t !.A/ :D ! .s, A/ds ,0 A 2 M . .t/ 0

If ! extends to a linear functional on M , we refer to it as a Dixmier trace.

If ! is a linear functional, then it is clear from the deﬁnition that it is a fully symmetric trace on M . The main result of this section is the following theorem on the existence of Dixmier traces.

Theorem 6.3.3. Let M be a Lorentz ideal of the algebra L.H / as above. Then M admits a Dixmier trace if and only if satisfies the condition (6.5).Indeed,ifM admits any nontrivial continuous trace then it admits a Dixmier trace. The end notes discuss how to adjust the definition of a Dixmier trace so that a similar result holds for a Lorentz operator space associated to an atomless semifinite von Neumann algebra. In fact, the definition and theorem above can be repeated verbatim for Lorentz operator ideals M .M, / in an arbitrary semifinite (non-finite) atom- 200 Chapter 6 Dixmier Traces less von Neumann algebra M. We emphasize the term ideal. Recall that the Lorentz operator ideal (of -compact operators in this case) is defined by

M .M, / :DfA 2 S.M, / : .A/ 2 M g, where satisfies one of the conditions in Remark 6.3.1. The ideals M .M, / are properly contained in the algebra M. For a Lorentz operator space on an arbitrary semifinite (non-finite) atomless von Neumann algebra that is not an ideal, an extended definition of a Dixmier trace is required. See the end notes. We prove Theorem 6.3.3. We define Dixmier traces on the Lorentz function space M so that we can use the lifting technique of Chapter 4. We say that the functional Z 1 t !.x/ :D ! .s, x/ds ,0 x 2 M .t/ 0 is a Dixmier trace on the Lorentz function space M if it extends to a linear functional on M . Theorem 4.4.1 indicates that a Dixmier trace exists on M if and only if a Dixmier trace exists on M . The following lemma shows that, for every satisfying the condition (6.5), there exists a suitable extended limit for the Dixmier trace construction.

Lemma 6.3.4. If : RC ! RC is a concave increasing function satisfying the condition (6.5), then there exists a dilation invariant extended limit ! on L1.0, 1/ such that t 0.t/ ! D 0. (6.6) .t/

Proof. Deﬁne a convex homogeneous functional : L1.0, 1/ ! R by the formula (6.3). Without loss of generality, we may assume that .0/ D 0 and, hence, by the Mean Value Theorem, there is some c 2 .0, t/ such that .t/=t D 0.c/ 0.t/. Hence, 0 t .t/= .t/, t>0, belongs to L1.0, 1/. By the condition (6.5), we have, for every T>1, .N T / .N T / lim inf D 1 and, therefore, inf D 1. N !1 .N / N>0 .N / Hence, Z NT d .s/ .N T / inf D inf log D 0. N>0 N .s/ N>0 .N / Therefore, Z 1 NT d .s/ .t 0.t/= .t// Dlim inf inf D 0. (6.7) T !1 N>0 log.T / N .s/ Section 6.3 Dixmier Traces on Lorentz Ideals 201

0 Let Y L1.0, 1/ be the real linear span of x0.t/ :Dt .t/= .t/ and 1. Deﬁne ! : Y ! R by setting

!.1 C 2x0/ D 1.

By (6.7), we have

1 C minf2,0g.x0/ 1 1 C maxf2,0g.x0/ or, equivalently,

.1 2x0/ !.1 C 2x0/ .1 C 2x0/.

Therefore, ! on Y . By the Hahn–Banach theorem, there exists a linear extension ! : L1.0, 1/ ! R such that ! lim sup . Therefore,

kxk1 .x/ !.x/ .x/ kxk1, x D x 2 L1.0, 1/.

Hence, ! 2 L1.0, 1/ . By Lemma 6.2.7, ! is a dilation invariant extended limit.

Lemma 6.3.5. Let ! be an extended limit on L1.0, 1/ and let x 2 L1.0, 1/ be such that !.x/ D a and x a. We have

(a) !.xy/ D !.x/!.y/ for all y 2 L1.0, 1/.

(b) !.1=x/ D 1=!.x/, if, in addition, 1=x 2 L1.0, 1/.

The same assertions are valid for extended limits on l1.

Proof. By the assumptions we have that

j!..x a/y/j!.jx ajkyk1/ Dkyk1!.x a/ D 0.

This proves the first assertion. The second assertion is a specific case of the first one by setting y D 1=x. The proof of the assertions for extended limits on l1 is identical, hence omitted.

Proof of Theorem 6.3.3. Let ! be an arbitrary extended limit on L1.0, 1/ and let 0 x, y 2 M . It follows from Theorem 3.3.3 that Z Z 1 t 1 t ! .s, x C y/ds ! ..s, x/ C .s, y//ds . .t/ 0 .t/ 0 Thus,

!.x C y/ !.x/ C !.y/. (6.8) 202 Chapter 6 Dixmier Traces

It follows from Theorem 3.3.4 that Z Z 1 t 1 2t ! ..s, x/ C .s, y//ds ! .s, x C y/ds . .t/ 0 .t/ 0 Therefore, Z .2t/ 1 2t !.x/ C !.y/ ! .s, x C y/ds . (6.9) .t/ .2t/ 0 Suppose now that the condition (6.5) holds. Select a dilation invariant extended limit ! satisfying the condition (6.6) (such a selection is possible by Lemma 6.3.4). It follows that .2t/ .t/ C t 0.t/ 1 ! ! D 1. .t/ .t/ It now follows from (6.9) and Lemma 6.3.5 that Z 1 2t !.x/ C !.y/ ! .s, x C y/ds . .2t/ 0 Since ! is dilation invariant, it follows that Z 1 t !.x/ C !.y/ ! .s, x C y/ds D !.x C y/. (6.10) .t/ 0

It follows from (6.8) and (6.10) that ! is an additive functional on the positive cone of M . Thus, it extends to a linear functional on M . Suppose that the condition (6.5) fails, then .t=n/ lim sup < 1 t!1 .t/ for every n 2 N.Bytheassumptionthat .t/ D O.t/ as t ! 0 it follows that .t=n/ sup D o.1/ t>0 .t/ as n !1. Thus,

˚n 0˚n 0 kx kM kxkM k kM DkxkM kn kM n .t=n/ DkxkM sup D o.n/ t>0 .t/ as n !1for every x 2 M . By Theorem 4.9.2, M does not admit any nontrivial symmetric functionals (including Dixmier traces). Section 6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces 203

We make an observation on (6.6). For certain choices of the function , the condition (6.6) can be satisﬁed by every dilation invariant extended limit on L1.0, 1/. For example, if .t/ :D log.1 C t/, t 0, then (6.6) is satisﬁed for every dilation invariant extended limit.

Theorem 6.3.6. Let M1,1 be the Lorentz ideal of compact operators associated to the function .t/ :D log.1 C t/, t 0.Then!, as in Deﬁnition 6.3.2,isaDixmier trace on M1,1 for every dilation invariant extended limit ! on L1.0, 1/.

Proof. We have t 0.t/ t 1 D D o.1/, t 0. .t/ t C 1 log.1 C t/

t 0.t/ If ! is a dilation invariant extended limit on L1.0, 1/,then!. .t/ / D 0, and (6.6) is satisﬁed. By the proof of Theorem 6.3.3 above, ! is a Dixmier trace.

6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces

In the last section we showed that a Dixmier trace exists on a Lorentz ideal M of compact operators if and only if a nontrivial continuous trace exists on M ,which occurs if and only if .2t/ lim inf D 1. t!1 .t/

Assume that satisfies this condition. Dixmier traces aprioriform a subset of the fully symmetric functionals on M . We prove in this section that every fully symmetric functional on the Lorentz ideal M of compact operators is a Dixmier trace up to a constant factor. This shows that Dixmier traces provide the construction of all fully symmetric functionals on Lorentz ideals of compact operators. Again, as in the preceding section, the result and the proof of Theorem 6.4.1 below can be repeated verbatim for Lorentz ideals M .M, /in an arbitrary semifinite (non-finite) atomless von Neumann algebra M.

Theorem 6.4.1. Suppose M is a Lorentz ideal of the algebra L.H / satisfying the condition (6.5). Then every fully symmetric functional on M is a Dixmier trace (up to a constant factor).

To prove the theorem, it is sufﬁcient to consider only the commutative case. By Corollary 4.4.2 there is a bijection between fully symmetric functionals on the operator ideal M and fully symmetric functionals on the function space M . Therefore, it 204 Chapter 6 Dixmier Traces

sufﬁces to show that a fully symmetric functional on M is, up to a constant, a Dixmier trace.

Recall that DM is the real linear span of all positive non-increasing functions from

M . Observe that x 2 DM if and only if x D .y/ .z/ for y, z 2 M . A fully 0 symmetric function ' on M is said to be normalized if '. / D 1. Since ' is fully symmetric this is equivalent to k'kM D 1.

Lemma 6.4.2. Let ' be a normalized fully symmetric functional on M . If x 2 DM , then Z 1 t '.x/ lim sup x.s/ds. (6.11) t!1 .t/ 0

Proof. Let x D .u/ .v/ with u, v 2 M . Denote the right-hand side of (6.11) by c.Foragiven>0, there exists T>0 such that Z Z t t x.s/ds .c C / 0.s/ds, t>T. 0 0 It follows that 0 .u/ .v/ C .c C / C .u/.0,T/.

Since ' is fully symmetric and must be singular, it follows that

0 '..u// '..v// C .c C /'. / C '..u/.0,T// D '..v// C c C .

Therefore, '.x/ c C .Since>0 is arbitrarily small, the assertion follows.

Now, we are able to prove the main result of this section.

Proof of Theorem 6.4.1. In what follows, ' is a normalized fully symmetric functional on M . To prove the result, we need to ﬁnd a dilation invariant extended limit !0 on

L1.0, 1/ such that !0 D '. Deﬁne a map T : DM ! L1.0, 1/ by the formula Z 1 t .T x/.t/ :D x.s/ds. .t/ 0

Since T is an injection, we can deﬁne a linear functional !0 on TDM by the formula !0.T x/ :D '.x/. By Lemma 6.4.2, we have

!0.T x/ D '.x/ lim sup.T x/.t/, x 2 DM . t!1 Section 6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces 205

Thus, !0 lim sup on TDM . By the Hahn–Banach theorem, there exists a linear extension !0 : L1.0, 1/ ! R such that !0 lim sup . That is, !0 is an extended limit. For every ﬁxed u>0, we have Z t 0 .ut/ D .u1=u /.s/ds. 0 It now follows from Lemma 4.2.4 that .ut/ ! D '.u 0/ D '. 0/ D 1. 0 .t/ 1=u By Lemma 6.3.5, we have Z Z 1 ut .t/ 1 t ! . .T x// D ! x.s/ds D ! .u x/.s/ds 0 1=u 0 .ut/ 0 .ut/ .t/ 1=u Z 0 0 1 t D !0 .u1=ux/.s/ds D '.u1=ux/ D '.x/ D !0.T x/ .t/ 0 for every x 2 DM .

Let Y be the linear span of u.TDM /, u>0. Observe that Y is a dilation invariant linear subspace of L1.0, 1/. It follows from the previous display that !0 is dilation invariant on Y . Thus, the subspace Y , the dilation semigroup s, s>0, the linear functional !0jY and the convex functional lim sup on L1.0, 1/ satisfy the conditions of Theorem 6.2.5. By Theorem 6.2.5, !0jY extends to a dilation invariant extended limit on L1.0, 1/. In Section 1.3.2 in Chapter 1 we deﬁned a Dixmier trace as follows.

Remark 6.4.3. If ! is a dilation invariant extended limit on l1, then the functional 1 Xn Tr! : A ! ! .k, A/ ,0 A 2 M .n C 1/ kD0 n 0 is also called a Dixmier trace (provided that it extends to a linear functional on M ). Let us reconcile this deﬁnition with Deﬁnition 6.3.2 so that Theorem 6.3.3 and Theorem 6.4.1 can be equally phrased in terms of Dixmier traces in the sense of Re- mark 6.4.3.

Theorem 6.4.4. Let M be a Lorentz ideal of the algebra L.H / satisfying the condition (6.5). Then the set of Dixmier traces !, ! a dilation invariant extended limit on L1.0, 1/, is identical to the set of Dixmier traces Tr!, ! is a dilation invariant extended limit on l1. 206 Chapter 6 Dixmier Traces

Proof. Suppose Tr! is a Dixmier trace for a dilation invariant extended limit ! on l1. Clearly, by construction, it is a fully symmetric functional on M . Hence, by 0 Theorem 6.4.1, there is some ! , an extended limit on L1.0, 1/, such that Tr! D !0 . Conversely, suppose ! is a Dixmier trace for a dilation invariant extended limit ! on L1.0, 1/. For any sequence a 2 l1,set X1 0 ! .a/ :D ! anŒn,nC1/.t/ . nD0

0 Then ! is a dilation invariant extended limit on l1. It follows from the fact that, for t 2 Œn, n C 1/ and A 2 M , Z 1 t 1 Xn .s, A/ds D .j , A/ C o.1/, .t/ .n/ 0 j D0 that we obtain 1 X 1 Xn ! .j , A/ C .t/ D .A/. .n/ Œn,n 1/ ! nD0 j D0

Hence, Tr!0 .A/ D !.A/ is linear and thus a Dixmier trace for the dilation invariant extended limit !0.

6.5 Dixmier Traces on Fully Symmetric Ideals of L.H /

In this section, we extend the concept of a Dixmier trace to an arbitrary fully symmetric ideal of L.H /. The main result of this section is Theorem 6.5.6, which indicates if and only if criteria for linearity of a Dixmier trace on a fully symmetric operator ideal. We will use these results in the next section to extend Theorem 6.4.1. We begin with a simple extension of states on L1.0, 1/.

Lemma 6.5.1. Every state ! on the algebra L1.0, 1/ admits an extension to an C additive mapping ! from the set L0 .0, 1/ of the positive measurable functions to RC [f1g. This extension is deﬁned by setting

C !.x/ :D supf!.y/ :0 y x, y 2 L1.0, 1/g,0 x 2 L0 .0, 1/. (6.12)

A formula analogous to (6.12) gives an extension of a state on l1 to an additive mapping on the set LC of all unbounded positive sequences. C Proof. Let x1, x2 2 L0 .0, 1/.If0 y 2 L1.0, 1/ is such that y x1 C x2,then there exist positive elements y1, y2 2 L1.0, 1/ such that y D y1 C y2, y1 x1 and Section 6.5 Dixmier Traces on Fully Symmetric Ideals of L.H / 207

y2 x2. It follows from (6.12) that

!.y/ D !.y1/ C !.y2/ !.x1/ C !.x2/.

Taking the supremum over all such y, we obtain

!.x1 C x2/ !.x1/ C !.x2/. (6.13)

Fix >0. Let yi 2 L1.0, 1/, i D 1, 2, be such that 0 yi xi and !.yi /> !.xi / . It follows from (6.12) that

!.x1/ C !.x2/ 2 C !.y1/ C !.y2/ D 2 C !.y1 C y2/ 2 C !.x1 C x2/.

Since is arbitrarily small, we obtain

!.x1/ C !.x2/ !.x1 C x2/. (6.14)

The assertion follows from (6.13) and (6.14).

C It follows directly from the formula (6.12) that the extension ! : L0 .0, 1/ ! R[f1g deﬁned in Lemma 6.5.1 is dilation invariant if and only if ! : L1.0, 1/ ! R is dilation invariant.

Lemma 6.5.2. For every state ! on the algebra L1.0, 1/ and for every x 2 C L0 .0, 1/, we have !.minfn, xg/ ! !.x/ as n !1.

Proof. Fix a sequence fxngn0 L1.0, 1/ such that xn x for every n 0, and such that !.xn/ ! !.x/ as n !1. Evidently, xn minfkxnk1, xgx. It follows that !.minfkxnk1, xg/ ! !.x/ as n !1.Ifkxnk1 !1, then we conclude the proof. If kxnk1 C for n 0, then !.x/ D !.minfn, xg/ for every n C and the assertion follows.

Lemma 6.5.3. Let ! be a state on the algebra L1.0, 1/. Let 0 z 2 L1.0, 1/ C be such that !.z/ D 0 and let u 2 L0 .0, 1/ be such that !.u/ < 1. It follows that !.uz/ D 0.

Proof. It is clear that u D minfn, ugC.un/C . It follows from Lemma 6.5.2 that the sequence !.minfn, ug/ converges to !.u/.Since!.u/ < 1, it follows that !..u n/C/ ! 0. On the other hand, we have uz nz Ckzk1.u n/C .Since!.z/ D 0, it follows that !.uz/ kzk1!..u n/C/. Taking the limit n !1, we conclude the proof. 208 Chapter 6 Dixmier Traces

C Lemma 6.5.4. Let ! be an extended limit on L1.0, 1/ and let ! : L0 .0, 1/ ! R C [f1gbe its extension as in Lemma 6.5.1. For every z 2 L0 .0, 1/, we have

!.z/ D 0 ” !.fzg/ D 0 8>0.

Proof. Let !.z/ D 0. For every >0, we have fzg z. It follows that !.fzg/ !.z/ D 0. Thus, !.fzg/ D 0. On the other hand, it follows from Lemma 6.5.2 that

!.z/ D lim !.minfz, ng/ lim !.fz<g/ C n!.fzg/. n!1 n!1

Since !.fzg/ D 0, it follows that

!.z/ !.fz<g/ .

Since >0 is arbitrarily small, the assertion follows.

The following deﬁnition extends Deﬁnition 6.3.2.

Deﬁnition 6.5.5. Let E be a fully symmetric ideal of the algebra L.H /.Foragiven concave increasing function : RC ! RC with M E and a given dilation invariant extended limit ! on L1.0, 1/, deﬁne a mapping ! : EC ! RC [f1gby setting Z 1 t !.A/ :D ! .s, A/ds ,0 A 2 E, .t/ 0

C where the extension of ! to L0 .0, 1/ is given by Lemma 6.5.1. If the mapping ! is ﬁnite and additive on EC, then its linear extension to E is called a Dixmier trace on E.

The following theorem gives a necessary and sufﬁcient condition for the mapping !, when ﬁnite, to be a trace on E.

Theorem 6.5.6. Let E be a fully symmetric ideal of the algebra L.H / and let M E. Let ! be a dilation invariant extended limit on the algebra L1.0, 1/ such that ! is ﬁnite on EC. The mapping ! is additive on EC if and only if

.2t/ !. / D 1. (6.15) .t/

Proof. Note that the convex function is subadditive and, therefore,

.2t/ t ! .t/ Section 6.6 Relatively Normal Functionals 209 is bounded. We may assume, without loss of generality, that : .n, n C 1/ ! R is a linear function for every n 2 R. Supposeﬁrst that ! is additive on EC. Select a positive operator A such that .A/ D 0 . Note that A 2 M E. It is obvious from the deﬁnition of A that .2t/ ! D .2 .A//. .t/ ! 1=2

The equality (6.15) now follows from Lemma 4.2.4. Assume now that the equality (6.15) holds. Let A, B 2 E be positive operators. It follows from Lemma 3.3.3 that

!.A C B/ !.A/ C !.B/. (6.16)

In order to prove the converse inequality, introduce the positive function z 2 L1.0, 1/ by setting .t/ z : t ! 1 , t>0. .2t/

By assumption, we have that !.z/ D 0. By Deﬁnition 6.5.5 and Theorem 3.3.4, we have Z 1 2t !.A/ C !.B/ ! .s, A C B/ds . .t/ 0 By Lemma 6.5.3, we have Z 1 2t .A/ C .B/ !..1 z.t// .s, A C B/ds/ ! ! .t/ 0 Z 1 2t D ! .s, A C B/ds . .2t/ 0 Since ! is dilation invariant, it follows that

!.A/ C !.B/ !.A C B/. (6.17)

The assertion now follows from (6.16) and (6.17).

6.6 Relatively Normal Functionals

In this section we extend Theorem 6.4.1. We prove that Dixmier traces, as deﬁned in the last section, are weak dense in the set of all fully symmetric functionals on a fully symmetric ideal E of compact operators that admits a nontrivial continuous trace. It is the main result of this chapter. 210 Chapter 6 Dixmier Traces

Theorem 6.6.1. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H /.IfE admits a nontrivial continuous trace, then there are Dixmier traces on E. Moreover, the set of those Dixmier traces is weak dense in the set of all fully symmetric traces on E.

The proof of the theorem has two stages. First, we introduce the class of “relatively normal functionals” (see Deﬁnition 6.6.4 below) and prove that they are weak dense in the set of all fully symmetric functionals (Theorem 6.6.5 below). Second, we prove that relatively normal functionals are Dixmier traces, in the sense of the previous section (Theorem 6.6.7 below). Let E be a fully symmetric ideal of compact operators and let ' be a fully symmetric functional on E. In what follows, EC denotes the positive cone of E. The next two lemmas permit us to introduce the notion of a relatively normal functional. By relatively normal we mean, more precisely, that the functional is normal with respect to a Lorentz ideal M E. There are many such ideals, in fact, every fully symmetric operator ideal is a union of Lorentz operator ideals M for a suitable set of concave functions , see e.g. [139]. Indeed, let A 2 E and let Z t A.t/ :D .s, A/ds. 0

It is easily checked that A is increasing, concave, and A.t/ D O.t/ as t # 0. M M M Evidently A 2 A .Furthermore,ifB 2 A ,thenB kBk A A.Bytheas- E E E M E sumption that is fully symmetric, then B 2 and kBk kBk A kAk . Clearly E M E M then, kBk D inf kBk A and D[ A where the inﬁmum and union are taken over all A 2 E with kAkE D 1.

Lemma 6.6.2. Let M E be a Lorentz ideal of compact operators and let ' be a fully symmetric functional on E. The mapping 'n, : EC ! R deﬁned by setting

'n, .A/ :D supf'.B/ : B 2 M ,0 B Ag,0 A 2 E, (6.18) is additive on the positive cone of E.

Proof. Let A1, A2 2 EC.LetB 2 M be such that 0 B A1 C A2.By [70, Theorem 2.2], there exists a linear operator C : L.H / ! L.H /,whichisa positive contraction both in L.H / and in L1, such that B D C.A1 C A2/. Setting B1 :D C.A1/ 0andB2 :D C.A2/ 0, we have B D B1 C B2. Therefore, 0 Bi B 2 M and Bi Ai , i D 1, 2. Hence, by the deﬁnition (6.18),

'.B/ D '.B1/ C '.B2/ 'n, .A1/ C 'n, .A2/. Section 6.6 Relatively Normal Functionals 211

Taking the supremum over all B satisfying the condition B A1 C A2, we obtain that

'n, .A1 C A2/ 'n, .A1/ C 'n, .A2/. (6.19)

Fix >0. There exist Bi 2 M such that 0 Bi Ai and '.Bi />'n, .Ai / . In particular, we have that

'n, .A1/ C 'n, .A2/ 2 C '.B1 C B2/. (6.20)

Further, we have that

B1 C B2 .B1/ C .B2/ .A1/ C .A2/ 21=2.A1 C A2/.

It follows from (6.20) and the deﬁnition (6.18) that

'n, .A1/ C 'n, .A2/ 2 C 'n, .21=2.A1 C A2// D 2 C 'n, .A1 C A2/.

Here, the last equality follows from Lemma 4.2.4. Since is arbitrarily small, we have

'n, .A1/ C 'n, .A2/ 'n, .A1 C A2/. (6.21)

The assertion follows from (6.19) and (6.21).

It is proved in the following lemma that 'n, can be viewed as the “normal part” of the functional ' with respect to the subspace M .

Lemma 6.6.3. The mapping 'n, : EC ! R extends to a fully symmetric functional on E. Moreover, ' D 'n, on M and .'n, /n, D 'n, on E.

Proof. Every additive functional on EC uniquely extends to a linear functional on E. In particular, so does 'n, : EC ! R. Let A1, A2 2 E be positive operators such that A2 A1. It follows that

f'.B/ : B 2 M ,0 B A2gf'.B/ : B 2 M ,0 B A1g.

Therefore, 'n, .A2/ 'n, .A1/. Hence, 'n, is a fully symmetric functional on E. The second assertion is obvious. In order to prove the third assertion, ﬁx a positive operator A 2 E. By deﬁnition, .'n, /n, .A/ 'n, .A/. Select Bj 2 M , j 1, such that 0 Bj A andsuchthat'.Bj / ! 'n, .A/. Clearly, 'n, .Bj / D '.Bj /. Thus, 'n, .Bj / ! 'n, .A/. Therefore, .'n, /n, .A/ 'n, .A/,andthe third assertion is proved.

Deﬁnition 6.6.4. A fully symmetric functional ' on E is called relatively normal if there exists a Lorentz ideal M E such that ' D 'n, on E. 212 Chapter 6 Dixmier Traces

Theorem 6.6.5. The set of relatively normal functionals on E is weak dense in the set of all fully symmetric functionals on E.

# E Proof.ThesetEC :Df.A/, A 2 g, equipped with the partial ordering given by # the Hardy–Littlewood submajorization, is a directed set. For every x 2 EC,let x 0 be a concave increasing function such that x D x. For every given fully symmetric E # functional ', consider the net f'n, x 2 , x 2 ECg. We claim that this net weak converges to the functional '. Recall that the base of the weak topology is formed by the sets

N.A1, :::, Aj , / :Df 2 E : j.Ak /j <,1 k j g.

Fix some neighborhood U of 0 in the weak topology. Select >0 and operators 0 Ak 2 E such that

N.A1, :::, Aj , / U . P j # Set y D kD1 .Ak/. It is clear that, for every x 2 EC such that y x,wehave M Ak 2 x . It follows from Lemma 6.6.3 that .'n, x '/.Ak / D 0. Therefore,

' 'n, x 2f 2 E : j.Ak /jD0, 1 k j gU .

We now prove that, if E admits a nontrivial continuous trace, then every relatively normal functional is a Dixmier trace in the sense of Deﬁnition 6.5.5. Let A, B be positive bounded operators. Let A^B be any positive bounded operator such that Z Z Z t t t .s, A ^ B/ds D min .s, A/ds, .s, B/ds , t>0. (6.22) 0 0 0

Lemma 6.6.6. Let E be a fully symmetric ideal of the algebra L.H / and let ' be a relatively normal functional on E. There exists a positive operator B 2 E such that

'.A/ D lim '.A ^ nB/,0 A 2 E. n!1

Proof. By the assumption that ' is relatively normal there exists a Lorentz ideal M E such that ' D 'n, . We may assume, without loss of generality, that : .j , j C 1/ ! R is a linear function for every j 0. Select a positive operator B such that 0 .B/ D . Observe that B 2 M E. For every positive A 2 E,wehave

'.A/ D 'n, .A/ D supf'.C/ : C 2 M ,0 C Ag

D lim supf'.C/ : kC kM j ,0 C Ag. j !1 Section 6.6 Relatively Normal Functionals 213

It now follows from the deﬁnition of the Lorentz ideal M that '.A/ D lim supf'.C/ : C jB,0 C Ag j !1 D lim supf'.C/ :0 C A ^ jBg. j !1 Since the functional ' is fully symmetric, it follows that '.A/ D lim '.A ^ jB/. j !1

By Theorem 4.1.3 and Theorem 6.6.5, if a fully symmetric ideal E 6D L1 of compact operators admits a nontrivial symmetric functional then E admits nontrivial relatively normal functionals.

Theorem 6.6.7. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H / that admits a nontrivial continuoustrace and let ' be a nontrivial relatively normal functional on E. There exists a Lorentz ideal M E such that Z 1 t '.A/ D ! .s, A/ds ,0 A 2 E .t/ 0

C for some dilation invariant extended limit ! on L1.0, 1/ (extended to L0 .0, 1/ by Lemma 6.5.1).

Proof. The functional 'jM is fully symmetric and we can choose such that ' is normalized. By Theorem 6.4.1, 'jM is a Dixmier trace. In particular, there exists a dilation invariant extended limit ! on L1.0, 1/ such that Z 1 t '.A/ D ! .s, A/ds ,0 A 2 M . (6.23) .t/ 0 E T C For every positive A 2 ,deﬁne .A/ 2 L0 .0, 1/ by setting Z 1 t T.A/ : t ! .s, A/ds. .t/ 0 We may assume, without loss of generality, that : .n, n C 1/ ! R is a linear function for every n 0. Select a positive operator B such that .B/ D 0. Observe that B 2 M E. Dividing the equality (6.22) by .t/, we obtain T.A ^ nB/ D minfT.A/, ng. It follows from (6.23) and Lemma 6.6.6 that

'.A/ D lim !.minfT.A/, ng/,0 A 2 E. n!1 By Lemma 6.5.2, we conclude that '.A/ D !.T.A// for every 0 A 2 E. The proof of Theorem 6.6.1 follows from Theorem 6.6.5 and Theorem 6.6.7. 214 Chapter 6 Dixmier Traces

6.7 Wodzicki Representation of Dixmier Traces

The previous section expressed every nontrivial relatively normal functional ' on a fully symmetric ideal E ¤ L1 of compact operators in the form of a Dixmier trace, Z 1 t '.A/ D ! .s, A/ds ,0 A 2 E (6.24) .t/ 0 R R C for some increasing concave function : C ! C and where ! : L0 .0, 1/ ! R [f0g is the extension of a dilation invariant extended limit ! : L1.0, 1/ ! R (as in Lemma 6.5.1). In [259], M. Wodzicki proposed constructing traces on ideals of compact operators by using a positive function : RC ! RC and applying some limiting procedure (Wodzicki used Stone–Cechˇ compactiﬁcation for this purpose) to the mapping 1 Xn A ! .k, A/ . .n C 1/ kD0 n 0 In this section, we prove that every relatively normal functional on a fully symmetric ideal of compact operators can be represented in the form proposed by Wodzicki. The main result of this section is Theorem 6.7.5. First let us convert (6.24) into the discrete form of a Dixmier trace. The following deﬁnition extends the one given in Remark 6.4.3.

Definition 6.7.1. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H /. For a given concave increasing function with M E and given dilation invariant extended limit ! on l1 define a mapping Tr! : EC ! RC [f1gby setting 1 Xn Tr .A/ :D ! .k, A/ ,0 A 2 E, ! .n C 1/ kD0 where the extension of ! to LC is given by Lemma 6.5.1. If the mapping Tr! is finite and additive on EC, then its linear extension to E is called a Dixmier trace on E. The proof of the following theorem is identical to that of Theorem 6.6.7, hence it is omitted.

Theorem 6.7.2. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H / and let ' be a nontrivial relatively normal functional on E. There exists a Lorentz ideal M E such that 1 Xn '.A/ D ! .k, A/ ,0 A 2 E .n C 1/ kD0 for some dilation invariant extended limit ! on l1 (extended to LC by Lemma 6.5.1). Section 6.7 Wodzicki Representation of Dixmier Traces 215

The Banach space l1 is a commutative C -algebra. Let ˇN be the set of all nontrivial homomorphic functionals on l1. Clearly, ˇN is a weak closed subset of the unitballofl1. By the Banach–Alaoglu theorem, the unit ball of l1 (and, therefore, the set ˇN) is weak compact. By the Gelfand–Naimark theorem, l1 is isometrically isomorphic (via the Gelfand transform) to the C -algebra of all continuous functions on ˇN.ThesetˇN is usually called the Stone–Cechˇ compactiﬁcation of N.Theset N1 D ˇNnN is frequently referred to as the set of all inﬁnite integers. Note that every element of N1 is an extended limit.

Lemma 6.7.3. The dilation semigroup n, n 1, acts on N1. Every dilation invariant extended limit admits a representation Z

!.x/ D x.p/d.p/, x 2 l1 N1 with being a ﬁnite regular dilation invariant Borel measure on N1.

Proof. Let ek Dfıkj gj 0, k 0, be the standard basis sequence in l1.Ifp 2 ˇN, then

p.ek/p.el / D p.ekel / D 0, k ¤ l.

In particular, at most one of the numbers p.ek/, k 0, is non-zero. Observe that N p 2 1 if and only if p.ek/ D 0forallk 0. For every p 2 ˇN and every n 2 N, the mapping x ! .nx/.p/ is a homomorphism. Hence, it corresponds to a point q 2 ˇN.Ifq … N1, then there exists k 0 such that q.ek/ ¤ 0. Thus,

.kCX1/n1 .kCX1/n1 p.ek/ D p ek D p.nek/ D q.ek/ ¤ 0. mDkn mDkn

Hence, p.em/ ¤ 0forsomekn m<.kC 1/n. Thus, p … N1. It follows that n acts on N1. By the Riesz–Markov theorem (see [188]), for every state ! on l1, there exists a ﬁnite regular Borel measure on ˇN such that Z !.x/ D x.p/d.p/. (6.25) ˇ N Kakutani and Nakamura noted in [122] that if the state ! is an extended limit, then the measure is supported on N1. For every x 2 l1,wehave Z Z 1 .! ı n/.x/ D .nx/.p/d.p/ D x.p/d. ı .n/ /.p/. (6.26) N1 N1 216 Chapter 6 Dixmier Traces

Since, by assumption, ! D ! ı n for all n 1, it follows from (6.25) and (6.26) that the measure is invariant with respect to the action of the dilation semigroup.

The extension of an extended limit to LC admits the same representation.

Corollary 6.7.4. Let ! be a dilation invariant extended limit. There exists a ﬁnite regular dilation invariant Borel measure on N1 such that Z !.x/ D x.p/d.p/ N1 for every x 2 LC.

Proof. Fix p 2 N1.Extendp to an additive functional on LC by Lemma 6.5.1. For every x 2 LC and for every n 2 N,wehave.minfx, ng/.p/ D minfx.p/, ng. For a given n 2 N, it follows from the above and from Lemma 6.7.3 that Z !.minfx, ng/ D minfx.p/, ngd.p/. N1

By Levi’s theorem, we have Z Z minfx.p/, ngd.p/ ! x.p/d.p/. N1 N1

The assertion now follows from Lemma 6.5.1.

As a result of this corollary every nontrivial relatively normal functional on a fully symmetric ideal can be written in the form proposed by Wodzicki.

Theorem 6.7.5. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H / and let ' be a nontrivial relatively normal functional on E. There exist a concave function and a ﬁnite regular dilation invariant Borel measure on N1 such that Z 1 Xn '.A/ D .k, A/ .p/d.p/,0 A 2 E. (6.27) N .n C 1/ 1 kD0

Proof. The assertion follows immediately from Theorem 6.7.2 and Corollary 6.7.4.

The next result shows that if a fully symmetric ideal admits any nontrivial continuous trace at all, then it admits a constructible trace in the form of a Dixmier trace.

Corollary 6.7.6. Let E ¤ L1 be a fully symmetric ideal of the algebra L.H /. The following conditions are equivalent. Section 6.8 Notes 217

(a) For some A 2 E, we have

1 ˚n kA kE 6! 0, as n !1. n

(b) The fully symmetric ideal E admits a nontrivial continuous trace ' 2 E .

(c) There exist a concave function and a ﬁnite regular dilation invariant Borel measure on N1 such that the mapping Z 1 Xn A ! .k, A/ .p/d.p/,0 A 2 E N .n C 1/ 1 kD0

extends to a linear functional on E.

Proof. By Theorem 4.1.3, E admits a nontrivial symmetric functional, condition (b), if and only if condition (a) holds. In the latter case, E must admit a nontrivial fully symmetric functional. By Theorem 6.6.5, the latter can be approximated by a relatively normal fully symmetric functional. By Theorem 6.7.5, a nontrivial relatively normal fully symmetric functional admits a representation (6.27), which is equivalent to (c).

6.8 Notes

Dixmier Traces The main theme of this chapter has its roots in the paper of J. Dixmier [62]. To give a reader some of the background to this work, we reprint parts of Dixmier’s letter to the conference “Singular Traces and their Applications” (Luminy, 2012, reprinted with permission and translated by V. Gayral).

“... In their second seminal article, Murray and von Neumann proved the existence of a trace on type II1 factors. They showed that this trace is essentially uniquely characterized by its purely algebraic properties. What happens for type I1 or II1 factors? In 1960, I had a clear notion of “normality” for positive linear forms and traces, and it was easy to see that a normal trace is essentially unique. Let us consider the case of type I1 factors, that is to say L.H /, H being a Hilbert space that we will suppose to be separable. Is the classical trace on L.H /C characterized by its algebraic properties? In other words, is a trace on L.H /C automatically normal? In 1950, I was convinced that the answer was yes (up to some trivial counter-examples). I didn’t solve this problem until 1965, after working on this question for 15 years—not every day, of course. I tackled this problem 4 or 5 times. Such perseveranceis quite common among researchers, but what was a bit exceptional here, is that at each attempt, I made progress. In 1955 (I do not guarantee the absolute exactness of the following dates) I had the ﬁrst idea of a counter-example: 218 Chapter 6 Dixmier Traces

for a positive compact operator A, with eigenvalues .0, A/ .1, A/ such that .n, A/ D O.1=.n C 1//,Ideﬁnedt.A/ D lim .n, A/=.1=.n C 1//, lim being an extended Banach limit. But I wasn’t able to prove additivity of t.A/. In 1960, during a mathematical dinner, I explained my problem to my neighbor Nachman Aronszajn. He suggested that I should replace .n, A/ by .0, A/ C .1, A/ CC.n, A/ and 1=.n C 1/ by 1 C 1=2 CC1=.n C 1/, say differently by log.n C 1/. The next day, I explored this possibility and I noticed that I was making progress. However I had, on the one hand to specify which Banach limit to use (this was not an issue as I had, a few years earlier, participated in the birth of amenable groups), and on the other hand to prove certain inequalities on eigenvalues; I was unable to do so. In 1965, I went back to the problem and succeeded in proving the inequalities. A few months later, I found out that those inequalities had been proved by Joseph Hersch. From then on, everything could be exposed within two pages and I wrote a note in Comptes Rendus de l’Académie des Sciences. A play by Marivaux is called "The Second Surprise of Love". In my case I expe- rienced "The Third Surprise of Singular Traces". The ﬁrst surprise was of course, my observation of their existence in 1965. Now, just like Alexandre Dumas does, let us jump through the years, to a 1985 lunch at IHES. I was talking with Alain Connes, and I told him my astonishment that singular traces don’t have application. In my mind, singular traces are pathological monsters and they should be useful to prove monstrous behavior of operators, representations. At that time, nobody (but me) knew about singular traces, Alain Connes was no exception. But as soon as he understood (after approximately thirty seconds), he told me: “this is what I need”. Indeed, a few weeks later, I saw a manuscript or a preprint (I don’t remember), which obtained from singular traces applications that were not monstrous at all. For me, this is the second surprise of singular traces. The third surprise requires us to fast forward again by 25 years: I discovered, with this meeting, that singular traces are now used in various domains of mathematics.”

The exposition given in Connes’ work [48] has inﬂuencedmany researchers. In this chapter, we mainly followed the line started in [67], which viewed Dixmier traces as a special case of fully symmetric functionals on Lorentz operator ideals. This line of thought was pursued further in [65, 66] and in [127], where it was established that all normalized fully symmetric functionals on Lorentz operator ideals are precisely Dixmier traces. This result is essentially our Theorem 6.4.1.

Dixmier Traces on Lorentz Operator Spaces of Atomless von Neumann Algebras If M is a von Neumann algebra equipped with a faithful semifinite normal trace and .1/ D 1, thenany Lorentz operator space M .M, /satisfying the condition (6.5) admits a Dixmier trace. That is, the “if” statement in Theorem 6.3.3 is true. Theorem 6.3.6 is true unchanged. The proof is given by the sufficient direction of the argument in Section 6.3. Further restrictions are required for the necessity of the condition (6.5) to the existence of Dixmier traces. If M is an atomless von Neumann algebra equipped with a faithful semifinite normal trace and .1/ D1,and .t/ D O.t/ as t # 0, then Theorem 6.3.3 and Theorem 6.4.1 are still true for the Lorentz operator ideal M .M, /, with exactly the same proofs as for the Lorentz Section 6.8 Notes 219

ideal of compact operators. If M is an atomless von Neumann algebra equipped with a faithful finite normal trace (without loss, .1/ D 1), then the condition .t/ D O.t/ as t # 0 means that M .M, / D M and, as such, the Lorentz operator ideals are of no interest and all Dixmier traces constructed from extended limits at 1 (as the extended limits in Section 6.2 are generally called) are trivial. To study Lorentz operator spaces associated to atomless semifinite von Neumann algebras in full generality (i.e. to drop the condition .t/ D O.t/ as t # 0andwhere may be finite or infinite), requires the notion of extended limits at 0 and singular symmetric functionals supported at 0 [67]. A state on the algebra L1.0, 1/ is an extended limit at 0 if it extends the functional limt#0 f.t/ on continuous functions. This theory is easy to understand in analogy with the extended limits at 1. Dixmier traces are defined as in Definition 6.3.2 using dilation invariant extended limits at 0 as well as dilation invariant extended limits at 1. With this notion of a Dixmier trace one can prove the same statements contained in Theorem 6.3.3 and Theorem 6.4.1 for an arbitrary Lorentz operator space associated to an atomless semifinite von Neumann algebra, provided that the condition lim inft#0 .2t/= .t/ D 1 is added to condition (6.5). Essentially all these results can be discerned in [127], see also [213].

Other Constructions of Traces The end notes to Chapter 1 indicated the contributions of Kalton [123], Figiel, Varga [245], and Pietsch [181,184] to the construction of traces on operator ideals. The constructions were independent of Dixmier. Wodzicki’s paper [259] suggested to transfer Dixmier’s construction to a much wider class of operatorideals. Questions of ﬁniteness and linearity become very hard in this general setting. The paper [238] generalizes some of the results from [259]. The techniques used in [259] and [238] are distinct. Theorem 6.5.6 is of special interest for fully symmetric (function) spaces which are not Lorentz spaces M and which still admit nontrivial fully symmetric functionals. For concrete examples of such spaces we refer the reader to [66].

Part III Traces on Lorentz Ideals

This part of the book discusses formulas for Dixmier traces on Lorentz operator ideals. In Chapter 6 we deﬁned a Dixmier trace ! with the formula Z 1 t !.A/ :D ! .s, A/ds ,0 A 2 M , .t/ 0 where ! is a dilation invariant extended limit on L1.0, 1/ such that the functional ! is additive. Here M is the Lorentz operator ideal M :D M .L.H /,Tr/ contained in the set of compact operators on the separable Hilbert space H , and the concave increasing function satisﬁes

.2t/ lim inf D 1. t!1 .t/

In Chapter 7 we give various Lidskii-type formulas for Dixmier traces. More precisely, we shall show in Theorem 7.3.2 that, for an arbitrary Dixmier trace !,wehave 1 X .A/ D ! , A 2 M . ! .t/ 2.A/,jj> .t/=t

On the right-hand side of the formula, ! should be treated as the extension to unbounded measurable functions of an extended limit, as in Lemma 6.5.1, and .A/ denotes the non-zero spectrum of the compact operator A, counted with multiplicities. Thus, this particular Lidskii-type formula represents a spectral counting formula for the Dixmier trace. The counting radius .t/=t can be improved in special cases. When ! is an M - invariant extended limit on L1.0, 1/,whereM is the logarithmic mean Z 1 t ds .M x/.t/ :D x.s/ , t>0, log.t/ 1 s and the Lorentz ideal is the Dixmier–Macaev ideal M1,1 of compact operators, we show in Theorem 7.4.3 that for every operator A 2 M1,1 we have 1 X .A/ D ! . ! log.1 C t/ 2.A/,jj>1=t 222 Part III Traces on Lorentz Ideals

This improvement from .t/=t to 1=t cannot be achieved in general. There are Dixmier traces ! on the Dixmier-Macaev ideal (with ! failing the condition of M - invariance) for which the formula immediately above fails. Chapter 8 considers formulations of the Dixmier trace as a heat kernel asymptotic formula and a (generalized) -function residue, as used in noncommutative geometry. Heat kernel and -function residue formulations of the Dixmier trace use distribution formulas and therefore, in Chapter 8 (and in Chapter 9), we provide formulas on the Dixmier–Macaev operator ideal of an arbitrary atomless or atomic (non-finite) semifinite von Neumann algebra .M, /. This is indicated in the chapter by using the explicit notation M1,1.M, /. Heat kernel and -function residue formulations involve the behavior of the functions 1 s ! .A1Cs/, s ! .exp.sA1 //, M t ! .exp..tA/1// , t where 0 A 2 M1,1.M, /, and the associated functionals 1 1 .A/ :D .A1C1=t/ , .A/ :D . ı M/ .exp..tA/1// t t where is an extended limit on L1.0, 1/. We study conditions on the extended limit under which the above defined functionals turn out to be Dixmier traces, and, using distribution formulas from Chapter 7 we provide definitive connections of the sets of traces given by and with Dixmier traces ! on M1,1.M, /. In particular, using the characterization in Chapter 6 of Dixmier traces as the set of all normalized fully symmetric functionals on a Lorentz operator ideal, we show in Theorem 8.3.6 that every Dixmier trace ! on M1,1.M, / coincides with some heat kernel functional ,where is dilation invariant. That is, the set of heat kernel formulas on M1,1.M, / coincides with the set of Dixmier traces on M1,1.M, /. Surprisingly, we show that a similar result between -function residues and Dixmier traces fails. The set of -function residues is strictly smaller than the set of Dixmier traces on M1,1.M, /. Although the set of heat kernel formulas and the set of Dixmier traces coincide, we warn that the direct correspondence, ! D ! , is not true in general. The correspondence holds when ! is an M -invariant extended limit, but there are dilation invariant extended limits (failing the condition of M -invariance) such that ! 6D ! . In Chapter 9, we consider Alain Connes’ notion of measurability of operators, which is when the value of the Dixmier trace of an operator is the same for all (or, for a selected subset of) Dixmier traces on a Lorentz operator ideal. This notion plays a central role in noncommutative geometry, as seen in Part IV. Differences in the counting radius for Lidskii formulas in Chapter 7, as mentioned above, and the connection between heat kernel formulas and Dixmier traces in Chapter 8, as mentioned above, Part III Traces on Lorentz Ideals 223 indicates that two sets of Dixmier traces play a special role: the set of all Dixmier traces ! and the subset generated by those ! which are M -invariant. Correspondingly, in Chapter 9, we consider the set of all Dixmier measurable operators within a Lorentz operator ideal (the value of the Dixmier trace of the operator is the same for all Dixmier traces) and the set of all M -measurable operators (the value of the Dixmier trace of the operator is the same for all Dixmier traces with M -invariant !). A remarkable fact presented in Theorems 9.3.1 and 9.5.1 is that these two classes admit rather different descriptions. The set of all M -measurable operators is properly larger than its counterpart of all Dixmier measurable operators (Theorem 9.5.4). Thus, while the Dixmier traces on a Lorentz operator ideal defined by M -invariant extended limits have slightly stronger Lidskii and heat kernel formulations, generally, it is not equivalent to restrict considerations to this smaller set of traces.

Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

7.1 Introduction

Recall (see Theorem 6.3.3) that a Lorentz ideal M of the algebra L.H / admits a nontrivial continuous singular trace if

.2t/ lim inf D 1, (7.1) t!1 .t/ in which case it admits Dixmier traces. A central feature in the study of Dixmier traces is Lidskii formulations. The general Lidskii formulation, Corollary 5.6.4 in Chapter 5, indicates that the Dixmier trace of a compact operator A is given by a symmetric functional applied to an eigenvalue sequence of A. However, the corollary does not specify the form of the latter symmetric functional. For the standard trace Tr on the symmetric ideal L1 of trace class operators, Lidskii’s theorem asserts that the trace has the formula X1 Tr.A/ D .n, T/ nD0 for any A 2 L1. Here, f.n, A/gn0 2 l1 is the sequence of eigenvalues of A, taken in anyP order. This arbitrariness of the order is due to the absolute convergence of the series n0 j.n, T/j. In particular, we can choose the decreasing order of absolute values of .n, T/, counting multiplicities. Our ﬁrst result is that Lidskii’s formula can be extended to Dixmier traces on a Lorentz ideal M of compact operators. That is, for any (discrete) Dixmier trace Tr! on M , 1 Xn Tr .A/ D ! .j , A/ , A 2 M . ! .1 C n/ j D0

Here .n, A/, n 0, is an eigenvalue sequence of A (see Deﬁnition 1.1.10), that is, the non-zero eigenvalues counting multiplicities ordered by decreasing absolute values, unless A is quasi-nilpotent and, in that case, the eigenvalue sequence is the zero sequence. 226 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

Our main result concerning Dixmier traces on a Lorentz ideal M of compact operators is given in Theorem 7.3.2, which asserts that, for any (continuous) Dixmier trace ! on M ,wehave 1 X .A/ D ! , A 2 M . (7.2) ! .t/ 2.A/,jj> .t/=t

Here .A/ is the non-zero spectrum of A counting multiplicities which, if empty, means that the sum is zero. On the right-hand side of this formula the extended dilation invariant state ! on L1.0, 1/ is extended to L0.0, 1/, as in Lemma 6.5.1. Our strategy in this chapter is the same as in Part II. We ﬁrst verify a commutative version of (7.2) on a Lorentz function space, which takes the form of spectral distribution formula, and then lift it to the corresponding Lorentz ideal of L.H /.

7.2 Distribution Formulas for Dixmier Traces

In this section, we prove spectral distribution formulas for a Dixmier trace on a Lorentz function, or sequence, space. They provide the commutative versions of the formula (7.2). Let : RC ! RC be a concave increasing function satisfying the condition (7.1). Throughout this chapter, M denotes the Lorentz function spaces from Exam- ple 2.6.10, Z 1 t M :D x 2 S.0, 1/ :sup .s, x/ds < 1 , t>0 .t/ 0 and 1 Xn m :D x 2 l1 :sup .j , x/ < 1 , n0 .n C 1/ j D0 denotes the Lorentz sequence spaces. Dixmier traces ! on the function space M (or the sequence space m ) are deﬁned as in Section 6.3. The following theorems are the principal results of this section.

Theorem 7.2.1. Let ! be a dilation invariant extended limit on L1.0, 1/ such that ! is a Dixmier trace on M . For every positive x 2 M , we have Z 1 1 !.x/ D ! dnx./ .t/ .t/=t where nx is the distribution function of x (see Example 2.3.9), and on the right-hand C side ! is extended to L0 .0, 1/ as in Lemma 6.5.1. Section 7.2 Distribution Formulas for Dixmier Traces 227

Proof. The assertion follows from Lemmas 7.2.6 and 7.2.8 below.

Theorem 7.2.2. Let ! be a dilation invariant extended limit on L1.0, 1/ such that ! is a Dixmier trace on m . For every positive x 2 m , we have 1 X .x/ D ! x.k/ . ! .t/ x.k/ .t/=t

C On the right-hand side ! is extended to L0 .0, 1/ as in Lemma 6.5.1.

Proof. Let ! be a Dixmier trace on m . By Theorem 4.4.4, ! extends to a symmetric functional ' on M . The extension is given by the formula '.x/ D !.E..x/jA//, 0 x 2 M (here, A Df.n, n C 1/gn0). It is obvious, however, that Z Z t t .s, x/ds D E..x/jA/.s/ds C O.1/. 0 0

It follows that ' D ! on M (with exactly the same !). The assertion now follows from Theorem 7.2.1.

We warn the reader that Theorems 7.2.1 and 7.2.2 would fail if we do not extend the C state ! to L0 .0, 1/. If we fail to extend !, the right-hand side may not make sense, as is shown by the following example.

Example 7.2.3. Let : RC ! RC be a concave increasing function such that

.2t/ lim sup D 2. (7.3) t!1 .t/

There exists 0 x 2 M such that the mapping Z 1 1 t ! dnx ./ (7.4) .t/ .t/=t is unbounded.

Proof. It follows from (7.3) that, for every n 2 N,

.nt/ lim sup D n t!1 .t/ or, equivalently, .t=n/ 1 lim inf D . t!1 .t/ n 228 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

The sequence tn, n 2 N, is constructed by the following inductive process. First, set t0 D 0andt1 D 1. If tn1 is already constructed, select tn so large that .tn/> 2 .tn1/ and such that .t =n/ 2 n < . (7.5) .tn/ n

Deﬁne the function x D .x/ 2 L1.0, 1/ by setting

2 .tn/ x.s/ :D , 8s 2 Œtn1, tn/. tn

We claim that x 2 M . Indeed, since .tn/>2 .tn1/, n 2, it follows that Z Z tn Xn tk Xn .s, x/ds D .s, x/ds 2 .tk/ 4 .tn/ 0 tk kD1 1 kD1 for every n 1. Thus, x 4 0 and the claim follows. 1 .t/ We claim that the mapping t ! t nx . t / is unbounded. Assume the contrary. .t/ N Let nx . t / nt for some n 2 and every t>0. It follows that .t/ x.nt/ , 8t>0. t It follows that

.t=n/ .tn=n/ x.tn 0/ D lim x.t/ lim D . (7.6) t#tn t#tn t=n tn=n On the other hand, it follows from (7.5) that

2 .tn/ .tn=n/ x.tn 0/ D > . (7.7) tn tn=n Clearly, (7.6) contradicts (7.7). This proves the claim. We prove Lemma 7.2.6 and Lemma 7.2.8. The following lemma is the important step; it also plays a role in the next section.

Lemma 7.2.4. Let ! be a dilation invariant extended limit on L1.0, 1/ such that is a Dixmier trace on M . For every positive x 2 M , we have ! t.t, x/ (a) ! D 0. .t/ 1 .t/ (b) ! n . / D 0, t x t C where, in this formula, ! is extended to L0 .0, 1/ as in Lemma 6.5.1. Section 7.2 Distribution Formulas for Dixmier Traces 229

Proof. First, we prove (a) and then derive (b) from it. (a) It follows from Lemma 4.2.4 that Z 1 1 t=2 !..x// D ! 2.x/ D ! .s, x/ds . 2 .t/ 0 Therefore, Z 1 1 t t.t, x/ 0 D !..x// ! 2.x/ D ! .s, x/ds ! . 2 .t/ t=2 2 .t/

The assertion follows immediately. (b) Applying Lemma 6.5.4 to the mapping z : t ! t.t, x/= .t/ and using Lem- ma 7.2.4 (a), we obtain

!.mft.t,x/= .t/1=mg/ D !.ft.t,x/= .t/1=mg/ D 0

for every m 2 N. On the other hand, we have

1 .t/ D .t/ D f.t=m,x/ .t/=tg f t nx . t /1=mg fnx . t /t=mg

D mf.t,x/ .mt/=mtg mf.t,x/ .t/=mtg

D mft.t,x/= .t/1=mg.

Hence, for every m 1, we have

!. 1 .t/ / D 0. f t nx . t /1=mg

Again applying Lemma 6.5.4, we obtain the assertion. The next lemma indicates (at least, on an intuitive level) that the summation order in (7.2) and in Theorems 7.2.1 and 7.2.2 is correct.

Lemma 7.2.5. Let : RC ! RC be a concave increasing function. For every 0 x 2 M , we have .t/ n kxk t, t>0. x M t

Proof. Recall that, for every x 2 M and every t>0, we have Z 1 t t.t, x/ kxkM .s, x/ds . .t/ 0 .t/ 230 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

It follows that .t/ .t, x/ kxk , t>0. (7.8) t M Since t ! .t/=t is a decreasing function, it follows that .t/ .s/ .t/ n kxk m s : D t. x M t s t The following lemma proves one direction of Theorem 7.2.1.

Lemma 7.2.6. Let ! be a dilation invariant extended limit on L1.0, 1/ such that ! is a Dixmier trace on M . For every positive x 2 M , we have Z 1 1 !.x/ ! dnx./ , .t/ .t/=t

C where, on the right-hand side, ! is extended to L0 .0, 1/. Proof. It follows from Lemma 7.2.5 that Z Z Z 1 nx.kxkM .t/=t/ t dnx ./ D .s, x/ds .s, x/ds. (7.9) kxkM .t/=t 0 0

If kxkM 1, then the assertion follows from (7.9) immediately. If kxkM 1, then Z Z 1 1 .t/ .t/ dnx./ dnx./CkxkM nx . (7.10) .t/=t kxkM .t/=t t t

It follows from (7.9) and (7.10) that Z 1 1 1 .t/ ! dnx./ !.x/ CkxkM ! nx . .t/ .t/=t t t By Lemma 7.2.4 (b), the second summand vanishes and the assertion follows.

Remark 7.2.7. Let : RC ! RC be a concave increasing function and let ! be a dilation invariant extended limit on L1.0, 1/ such that ! is a Dixmier trace on . We have M .nt/ ! D 1 .t/ for every n 1. Indeed, it follows from Lemma 4.2.4 that .nt/ ! D .n 0/ D . 0/ D 1. .t/ ! 1=n ! Section 7.2 Distribution Formulas for Dixmier Traces 231

The lemma below establishes the other direction in Theorem 7.2.1.

Lemma 7.2.8. Let ! be a dilation invariant extended limit on L1.0, 1/ such that ! is a Dixmier trace on M . For every positive x 2 M , we have Z 1 1 !.x/ ! dnx./ , (7.11) .t/ .t/=t

C where, on the right-hand side, ! is extended to L0 .0, 1/.

Proof. Fixing n 2 N and writing Z Z 1 t .nt/ 1 t .s, x/ds D .s, x/ds .t/ 0 .t/ .nt/ 0 we obtain (via Remark 7.2.7 and Remark 6.3.5) the following formula for the left-hand side of (7.11) Z 1 t !.x/ D ! .s, x/ds . (7.12) .nt/ 0

We claim that Z Z t nx. .nt/=nt/ 1 .s, x/ds .s, x/ds C .nt/. 0 0 n

The inequality is evident if t nx. .nt/=nt/.Ift>nx . .nt/=nt/,then.s, x/ .nt/=nt for every s 2 Œnx . .nt/=nt/, t. Thus, Z Z Z t nx . .nt/=nt/ t .s, x/ds D .s, x/ds C .s, x/ds 0 0Z nx . .nt/=nt/ nx. .nt/=nt/ .nt/ .nt/ .s, x/ds C t nx 0 nt nt and the claim follows. Hence, Z Z 1 t 1 nx . .nt/=nt/ 1 ! .s, x/ds ! .s, x/ds C . .nt/ 0 .nt/ 0 n

It follows from (7.12) and the dilation invariance of ! that Z 1 nx . .t/=t/ 1 !.x/ ! .s, x/ds C . .t/ 0 n 232 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

Since n is arbitrary large, it follows that Z 1 nx . .t/=t/ !.x/ ! .s, x/ds . .t/ 0 Clearly, the right-hand side of the latter inequality coincides with that of (7.11). This proves the assertion.

7.3 Lidskii Formulas for Dixmier Traces

In this section we prove Lidskii formulas for Dixmier traces, as deﬁned in Chapter 6, on Lorentz ideals of the algebra L.H /.Let : RC ! RC be a concave increasing function satisfying the condition (7.1). If A 2 M has a Jordan decomposition

A D A1 A2 C iA3 iA4,0 A1, A2, A3, A4 2 M , then it is immediate from the deﬁnition of a Dixmier trace that 1 Xn Tr .A/ D ! ..k, A / .k, A / C i.k, A / i.k, A // . ! .n C 1/ 1 2 3 4 kD0

However, in this section we are interested in the spectral form of the Dixmier trace. Below, we express the Dixmier trace in terms of eigenvalues rather than singular values. The ﬁrst Lidskii formulation follows from Chapter 5. It is the simplest extension of Lidskii’s formula to Dixmier traces.

Theorem 7.3.1. For every dilation invariant extended limit ! on l1 such that Tr! is a Dixmier trace on M , we have 1 Xn Tr .A/ D ! .j , A/ , A 2 M , ! .n C 1/ j D0 where .A/ 2 m is an eigenvalue sequence of A. The proof of Theorem 7.3.1 is given below. The formula in Theorem 7.3.1 involves counting over the ﬁrst n non-zero eigenvalues, which are ordered so that their absolute value is decreasing. This ordering is not necessarily unique and the formula above holds for any such ordering. The next Lidskii formulation is a spectral counting version, it counts instead over the eigenvalues with absolute value greater than .t/=t. This theorem, establishing formula (7.2), is the main result of this chapter. Section 7.3 Lidskii Formulas for Dixmier Traces 233

Theorem 7.3.2. For every dilation invariant extended limit ! on L1.0, 1/ such that ! is a Dixmier trace on M , we have 1 X .A/ D ! , A 2 M , (7.13) ! .t/ 2.A/,jj> .t/=t where .A/ is the non-zero spectrum of A counting multiplicities, and on the right- hand side ! is extended to L0.0, 1/ as in Lemma 6.5.1. We now prove both theorems.

Proof of Theorem 7.3.1. Let A 2 M be self-adjoint. We have 1 Xn Tr .A/ D Tr .AC/ Tr .A/ D ! ..k, AC/ .k, A// . ! ! ! .n C 1/ kD0 It follows from Lemma 5.2.7 that ˇ ˇ ˇ Xn ˇ ˇ ˇ ˇ ..k, A/ .k, AC/ C .k, A// ˇ 2.n C 1/.n, A/. kD0 By Lemma 7.2.4 (a), we have 1 Xn Tr .A/ D ! .k, A/ . ! .n C 1/ kD0

Let A 2 M be normal. Using the previous paragraph, we have 1 Xn Tr .A/ D Tr .

The assertion follows for arbitrary A 2 M by using Theorem 5.5.1, since A D N C Q where N 2 M is normal, Tr!.Q/ D 0and.A/ D .N /. 234 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

The proof of Theorem 7.3.2 uses the distribution form of the Dixmier trace in The- orem 7.2.2 of the last section, and an additional technical lemma.

Lemma 7.3.3. If A 2 M is a normal operator, then 1 X 1 X (a) ! < D ! . .t/ .t/ 2.A/,jj> .t/=t 2. .t/=t 1 X 1 X (b) ! = D ! . .t/ .t/ 2.A/,jj> .t/=t 2.=A/,jj> .t/=t

Proof. Since A is a normal operator, it follows from Lemma 5.2.10 and Remark 2.3.8 that ˇ ˇ ˇ nA.X .t/=t/ ˇ ˇ ˇ .t/ .t/ ˇ ..k, A/ .k,

On the other hand, normality implies that j .t/=t kD0

Combining (7.14) and (7.15), we obtain ˇ ˇ ˇ X X ˇ ˇ ˇ .t/ .t/ ˇ < ˇ 6 1 C nA . t t 2.A/,jj> .t/=t 2. .t/=t

The ﬁrst assertion follows by Lemma 7.2.4 (b). The proof of the second assertion is identical.

Proof of Theorem 7.3.2. Denote, for brevity, the right-hand side of (7.13) by Lid!.A/. If A 2 M is a self-adjoint operator, then it follows from Theorem 7.2.2 that

Lid!.A/ D Lid!.AC/ Lid!.A/ D !.AC/ !.A/ D !.A/. Section 7.4 Special Cases and Counterexamples 235

If A 2 M is a normal operator, then one can read the assertion of Lemma 7.3.3 as follows

Therefore,

Lid!.A/ D Lid!.

The assertion follows for an arbitrary operator A 2 M by using Theorem 5.5.1, since A D N C Q where N 2 M is normal, !.Q/ D 0and.A/ D .N/.

7.4 Special Cases and Counterexamples

ThelogarithmicmeanM : L1.0, 1/ ! L1.0, 1/ is given by Z 1 t ds .M x/.t/ D x.s/ , t>0. log.t/ 1 s

In the special case when the Lorentz ideal is the Dixmier–Macaev ideal M1,1 of compact operators, and the extended limit ! is M -invariant1 (that is, ! D ! ı M ), one can improve the formula in Theorem 7.3.2. First let us indicate that every M -invariant extended limit on L1.0, 1/ is a dilation invariant extended limit. Then, as a consequence of Theorem 6.3.6, to every M -invariant extended limit on L1.0, 1/ we can associate a Dixmier trace ! on M1,1. The following lemma can be found in [48].

Lemma 7.4.1. For every extended limit ! on L1.0, 1/ the state ! ı M is a dilation invariant extended limit on L1.0, 1/.

Proof. Clearly, M maps positive functions into positive functions and M.1/ D 1. Therefore, ! ı M is a state. For every x 2 L1.0, 1/,wehave

lim x.t/ D 0 H) lim .M x/.t/ D 0. t!1 t!1

Hence, for every such x,wehave.! ı M /.x/ D 0. Therefore, ! ı M is an extended limit.

1 The existence of M -invariant extended limits on L1.0, 1/ can be shown in the same fashion as the existence of dilation invariant extended limits, as done in Section 6.2. The Hahn–Banach theorem, n Theorem 6.2.5, can be used on the action of N on L1.0, 1/ given by M , n 1. 236 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

For every x 2 L1.0, 1/,wehave ˇ Z Z ˇ ˇ t=s t ˇ 1 ˇ du duˇ j.Ms x M x/.t/jD ˇ x.u/ x.u/ ˇ j log.t/j u u ˇ Z1=s Z 1 ˇ ˇ 1 t ˇ 1 ˇ du duˇ 2kxk1 D ˇ x.u/ x.u/ ˇ . j log.t/j 1=s u t=s u j log.t/j

Therefore, !.Ms x Mx/ D 0. Equivalently, ! ı M is dilation invariant. The following lemma shows that every M -invariant state is approximately invariant with respect to the exponentiation semigroup. The action of the exponentiation semi- s group is notated by Ps : L1.0, 1/ ! L1.0, 1/, s>0, where .Ps x/.t/ D x.t /, x 2 L1.0, 1/, s, t>0.

Lemma 7.4.2. Let ! be an extended limit on L1.0, 1/. If ! D !ıM , then !ıPs ! ! in the weak topology as s ! 1.

Proof. It is clear that Z Z t t s 1 s du 1 du ..M ı Ps/x/.t/ D x.u / D x.u/ D .Ps .M x//.t/ log.t/ 1 u s log.t/ 1 u and

ˇ Z s ˇ ˇ t ˇ ˇ 1 duˇ j.Ps .M x//.t/ .M x/.t/jDˇ x.u/ ˇ kxk1 log.s/. log.t/ t u Since ! is M -invariant and s ! 1, it follows that

.! ı Ps /.x/ D .! ı M /.Ps x/ D .! ı Ps /.M x/ ! !.Mx/ D !.x/.

In the next theorem, unlike Theorem 7.3.2, we do not need to consider the extension of ! to unbounded measurable functions. The function 1 X t ! log.1 C t/ 2.A/,jj>1=t is bounded when A 2 M1,1.

Theorem 7.4.3. Let ! be an extended limit on L1.0, 1/ such that ! D ! ı M .For every operator A 2 M1,1, we have 1 X .A/ D ! ! log.1 C t/ 2.A/,jj>1=t Section 7.4 Special Cases and Counterexamples 237 where .A/ is the non-zero spectrum of A counting multiplicities. The same assertion holds if, alternatively, ! D ! ı Ps, s>0.

Proof. We prove only the commutative analog of this theorem. One derives the noncommutative result using the argument in Theorem 7.3.2. Let x 2 M1,1 be a positive function. We claim that Z Z t nx .1=t/ .s, x/ds .s, x/ds C 1, t>0. 0 0

Indeed, if t nx .1=t/, then the claim follows trivially. If t>nx .1=t/,then.s, x/ 1=t for every s 2 .nx.1=t/, t. It follows that Z Z Z t nx .1=t/ 1 1 nx.1=t/ .s, x/ds .s, x/ds C t nx .s, x/ds C 1. 0 0 t t 0

The claim is proved. It follows that Z 1 nx .1=t/ !.x/ ! .s, x/ds . log.1 C t/ 0

1C In order to prove the converse inequality, note that nx .1=t/ D O.t / (see also Lemma 8.2.8) for every >0. It follows that, for all sufﬁciently large t,wehave

Z Z 1C 1 nx .1=t/ 1 t .s, x/ds .s, x/ds log.1 C t/ 0 log.1 C t/ 0 Z 1C 1 C t 1C .s, x/ds. log.1 C t / 0

It follows that Z Z 1 nx.1=t/ 1 t ! .s, x/ds .1C/.!ıP1C/ .s, x/ds . log.1 C t/ 0 log.1 C t/ 0

If ! is Ps-invariant, s>0, then the assertion is shown. If ! is M -invariant, then the assertion follows from Lemma 7.4.2.

We now show that there are Dixmier traces ! on M1,1 (where ! is not M - invariant) for which Theorem 7.4.3 fails. To this end we use the extended limit ! provided by the lemma below. 238 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

Lemma 7.4.4. There exists a dilation invariant extended limit ! on L1.0, 1/ such that [ 2k kC2k !.A/ D 1, A D Œ2 ,2 /. k1

Proof. Deﬁne the convex homogeneous functional by the formula (6.3). That is, Z 1 NT x.s/ds .x/ D lim sup sup , x 2 L1.0, 1/. (7.16) T !1 N>0 log.T / N s

k Setting N D 22 and T D log.N / D 2k log.2/, we infer from (7.16) that

k Z kC2 1 2 log.2/ x.s/ds .x/ lim sup , x 2 L .0, 1/. k 1 k!1 k log.2/ 22 s

In particular, 1 .A/ 1. Thus, .A/ D 1. By the Hahn–Banach theorem, there exists a linear functional ! on L1.0, 1/ such that ! and !.A/ D 1. The assertion follows from Lemma 6.2.7.

Deﬁne a function x by the formula

2k k x :D sup 2 kC2 . (7.17) k1 Œ0,2

k1 k Fix k 1. For every t 2 Œ2k1C2 ,2kC2 ,wehave

Z Z k kC2 Xk t 2 n n x.s/ds x.s/ds 22 2nC2 2kC1 4log.1 C t/, 0 0 nD1 which guarantees that x 2 M1,1.

Lemma 7.4.5. Let x be as in (7.17) and let ! be as in Lemma 7.4.4. We have 1 .x/ D . ! log.2/

Proof. If [ k k A D Œ22 ,2kC2 /, k1 then it follows from Lemma 7.4.4 that Z A.t/ t !.x/ D ! .s, x/ds . (7.18) log.1 C t/ 0 Section 7.4 Special Cases and Counterexamples 239

k k For t 2 A, there exists k 1 such that t 2 Œ22 ,2kC2 /.Wehave

Z Z k1 t 2k1C2 .s, x/ds D .s, x/ds C t.t, x/ C O.1/. 0 0 On the other hand, we have

Z k1 2k1C2 Xk1 n n n1 .s, x/ds D 22 .2nC2 2n1C2 / D 2k C O.1/. 0 nD1

Therefore, Z t log.1 C t/ .s, x/ds D .1 C o.1// C t.t, x/ C O.1/, t 2 A. 0 log.2/ It now follows from (7.18) that 1 t.t, x/ .x/ D !.A/ C ! A.t/ . ! log.2/ log.1 C t/

By Lemma 7.2.4, the second term on the right-hand side is 0. The assertion now follows from Lemma 7.4.4.

Lemma 7.4.6. Let x be as in (7.17) and let ! be as in Lemma 7.4.4. We have Z 1 2 ! x.u/du D . log.1 C t/ x>1=t log.2/

Proof.If [ k k A D Œ22 ,2kC2 /, k1 then it follows from Lemma 7.4.4 that Z Z 1 A.t/ ! x.u/du D ! x.u/du . (7.19) log.1 C t/ x>1=t log.1 C t/ x>1=t 240 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

k k For t 2 A, there exists k 2 N such that t 2 Œ22 ,2kC2 /.Wehave

Z Z k 2kC2 x.u/du D x.u/du x>1=t 0 Xk n n n1 D 22 .2nC2 2n1C2 / nD1 k X log.1 C t/ D O.1/ C 2k D .2 C o.1//. log.2/ nD1

It now follows from (7.19) and Lemma 7.4.4 that Z 1 2 2 ! x.u/du D !.A/ D . log.1 C t/ x>1=t log.2/ log.2/

We provide a counterexample to Theorem 7.4.3 when ! 6D ! ı M ,aspromised. The same proof yields a counterexample for the atomless case, which we make use of in Chapter 8.

Theorem 7.4.7. There exists a positive operator A 2 M1,1 and a Dixmier trace ! such that 1 X .A/ ¤ ! . (7.20) ! log.1 C t/ jj>1=t,2.A/ The same statement holds for the atomless case, that is, there exist a positive operator A 2 M1,1.M, / and a Dixmier trace ! such that Z 1 1 !.A/ ¤ ! dnA./ . (7.21) log.1 C t/ 1=t

Here M is an atomless von Neumann algebra equipped with a faithful normal semiﬁ- nite trace .

Proof. Let x be as in (7.17) and let ! be as in Lemma 7.4.4. Let A be a positive operator such that .A/ D x. The assertion (in both cases) follows from Lemma 7.4.5 and Lemma 7.4.6. Section 7.5 Diagonal Formulas for Dixmier Traces Fail 241

7.5 Diagonal Formulas for Dixmier Traces Fail L If fekgk0 is an orthonormal basis of a Hilbert space H , then, for every A 2 1,we have X1 Tr.A/ D hAek, eki. kD0

This formula does not depend on which orthonormal basis is chosen. In this section we show that, in sharp contrast with Lidskii’s formula, the analog of the above formula fails for Dixmier traces.

L Lemma 7.5.1. Let A 2 .H / be a positive operator and let fekgk0 be an orthonormal basis of H . It follows that fhAek, ekigk0 .A/.

Proof. Without loss of generality, hAel , el ihAek , eki for l k.Letpn be the n projection onto the linear span of fekgkD0. It follows from Lemma 3.3.2 that Xn Xn hAek, ekiDTr.pnApn/ .k, A/. kD0 kD0

If E is a fully symmetric ideal of compact operators associated to a fully symmetric sequence space E, then Lemma 7.5.1 says that fhAek, ekigk0 2 E for any A 2 E E and any orthonormal basis fekgk0. Consequently, diag.fhAek, ekigk0/ 2 for any E A 2 and any orthonormal basis fekgk0, where diag denotes the diagonal operator with respect to any orthonormal basis of H . If ! is a Dixmier trace on a Lorentz ideal M of compact operators, then we are interested in whether the formula

M !.A/ D ! ı diag.fhAek, ekigk0/, A 2 , (7.22) M can hold for every A 2 and every orthonormal basis fekgk0 of H .Fromthe Lidskii formula for a Dixmier trace in Theorem 7.3.1 we know that the formula (7.22) holds for some operators and some orthonormal bases, for example, take A self-adjoint with trivial kernel and an appropriately ordered basis of eigenvectors for A. In fact, the formula (7.22) for selected operators and bases will be an important feature of Part IV. The crucial component in proving that such a formula cannot hold for every operator on M and every orthonormal basis is the following theorem proved by Victor Kaftal and Gary Weiss [120].

Theorem 7.5.2. Let 0 x, y 2 c0 be such that y x and y … l1. There exists a L positive compact operator A 2 .H / and some orthonormal basis fekgk0 such that .A/ D .x/ and y DfhAek, ekigk0. 242 Chapter 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals

Corollary 7.5.3. Let E be a fully symmetric ideal of compact operators and let ' be a fully symmetric singular trace on E. For any positive operator A 2 E, there exists an orthonormal basis fekgk0 of H such that

' ı diag.fhAek, ekigk0/ D 0.

In particular, for a Dixmier trace ! and a positive operator A 2 M ,formula(7.22) holds for every orthonormal basis if and only if !.A/ D 0.

Proof. Let be (the concave majorant of) the fundamental function of E.Thatis, M E M .n/ DkŒ0,nkE , n 0. It is known (see e.g. [139]) that .Since is M separable, it follows that every element of belongs to the closure (in the norm E L M topology of )oftheideal 1.Since' is singular, it follows that ' vanishes on . Let A 2 E be a positive operator. If A 2 L1,then'.A/ D 0 and the assertion follows from Lemma 7.5.1. From now on, we assume that A 62 L1. E L M L M L If ¤ 1,then ¤ 1. Fix an operator B 2 such that B … 1.Deﬁnea concave increasing function : RC ! RC by setting Z Z t t .t/ :D min .s, A/ds, .s, B/ds , t>0. 0 0

0 Set y D 2 L1.0, 1/.Sincey is constant on every interval .n, n C 1/, n 0, we obtain y 2 l1. Observe that y .A/ and y … l1. By Theorem 7.5.2, y D fhAek, ekigk0 for some orthonormal basis fekgk0.Sincey .B/, and since ' is fully symmetric and '.B/ D 0, it follows from Theorem 4.4.1 that ' ıdiag.y/ D 0. The ﬁrst assertion is shown. Suppose A 2 M is positive such that !.A/ D 0. By Lemma 7.5.1 and the fact that a Dixmier trace is a singular fully symmetric trace on M ,wehavethat0 ! ıdiag.fhAek, ekigk0/ !.A/ D 0. Formula (7.22) therefore holds. Conversely, if (7.22) holds for every orthonormal basis, then the ﬁrst assertion proves that !.A/ D 0. The second assertion is shown.

7.6 Notes

Lidskii Formula The material in this chapter may be viewed as an attempt to extend the classical diagonal and Lidskii formulas to Dixmier traces. Detailed reference to Lidskii’s original 1959 result can be found in the notes to Chapter 1. The second author’s interest in this theme began with a meeting with Thierry Fack in Delft in 1998, at which, during informal discussions, Thierry stated the following important question: “is every Dixmier trace spectral?” The answer to this question was (independently and almost simultaneously) given in the papers [89] and [81,82]. A proposition in [89] noted the formula in Theorem 7.3.1 for a classical pseudo-differential operator of order d on a closed d -dimensional Riemannian manifold, a topic we return to Section 7.6 Notes 243

in Part IV. However, generally, the question how precisely to compute the value of a given Dixmier trace from the spectrum of a given operator remained open. The Lidskii formulas as shown in this chapter were noted in [6] under signiﬁcant additional constraints on ! (for T 2 L1,1).

Theorem (from [6]). Let ! be M -invariant and let T 2 L1,1. We have 1 X .T / D ! , ! log.n/ jj>1=n,2.T/

where .T/ is the spectrum of T .

In the case when T is a positive arbitrary element from M1,1 and ! is taken from a rather special subset of all M -invariant extendedlimits (termed in [29] “maximallyinvariant Dixmier functionals”), this result can already be found in [32, Proposition 2.4]. In [9, Theorem 1], the assertion from [32, Proposition 2.4] was extended to an arbitrary M -invariant !. Another modiﬁcation of the class of extended limits for which the results [32, Proposition 2.4] and [9, Theorem 1] hold is given in [33, Proposition 4.3]. The spectral counting formula in the form of the radius .n/=n was achieved in [215]. The approach in [215] forms the basis of this chapter. Chapter 8 Heat Kernel Formulas and -function Residues

8.1 Introduction

The interplay between Dixmier traces, -function residues and heat kernel formulas is a cornerstone of noncommutative geometry [48]. These formulas are widely used in physical applications. To define these objects, fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace . Let A and B be positive operators from M. Consider the following Œ0, 1-valued -functions s ! .A1Cs/, s ! .A1CsB/. (8.1)

Clearly, these mappings are defined for s D 0 if and only if A 2 L1.M, /.This 1Cs 1 chapter shows (see Lemma 8.6.2) that .A / D O.s / if A 2 M1,1.M, /,where M1,1.M, /is the Dixmier–Macaev operator ideal associated to the pair .M, /,see Example 2.6.10. It therefore makes sense to ask about a residue of the so defined - function at s D 0. The latter can be defined using extended limits by setting 1 1 .A/ :D .A1C1=t/ , .A/ :D .A1C1=tB/ , (8.2) t ,B t where is an extended limit on L1.0, 1/. Consider the following Œ0, 1-valued heat kernel functions. Set

s ! .exp.sA1//, s ! .exp.sA1/B/. (8.3)

1 1 This chapter shows that .exp.sA // D O.s / if A 2 L1,1.M, /(here we recall that L1,1.M, / is the smaller operator ideal inside the Dixmier–Macaev operator ideal, consisting of those operators A 2 M such that .t, A/ D O.t1/). It is also shown that the functions 1 1 M t ! .exp..tA/1// , M t ! .exp..tA/1//B (8.4) t t belong to L1.0, 1/ if A 2 M1,1.M, /.HereM is the logarithmic mean. It therefore makes sense to ask about a residue of the so deﬁned heat kernel function at s D 0. Section 8.1 Introduction 245

The latter can be deﬁned using extended limits by setting 1 1 .A/ :D . ı M/ .exp..tA/ // , (8.5) t 1 .A/ :D . ı M/ .exp..tA/1//B , (8.6) ,B t where is an extended limit on L1.0, 1/. We prove in Section 8.2 that if ! is a dilation invariant extended limit, then the heat kernel functional ! is a linear functional on M1,1.M, /. In fact, we show in Proposition 8.2.5 that if ! is any extended limit such that ! is linear on M1,1.M, /, then necessarily there exists a dilation invariant extended limit !0 such that ! D !0 . Moulay-Tahar Benameur and Thierry Fack, in [9], suggested a more general approach to heat kernel formulas. It consisted of replacing the function t ! exp.t 1/ with an arbitrary Schwartz function f . The following equality was proved in [9], Z 1 1 1 ! .f.tA/B/ D f ds ! .AB/ (8.7) t 0 s for A 2 L1,1.M, /under the assumption that ! is M -invariant (that is, ! D !ıM ). In Theorem 8.5.1 below, we show that (8.7) holds for any dilation invariant extended limit !, for any twice-differential bounded function f on the semi-axis such that 0 f.0/ D f .0/ D 0, and for any A 2 M1,1.M, /. The main result of this chapter is the connection of heat kernel formulas and - function residues to Dixmier traces. We recall, from the methods of Chapter 6, that a Dixmier trace ! can be deﬁned on the Dixmier–Macaev operator ideal M1,1.M, / by lifting, as in Section 4.4, a Dixmier trace from the Lorentz function space M1,1, and that the set of Dixmier traces coincides with the set of all normalized fully symmetric functionals on M1,1.M, /.Here! is any dilation invariant extended limit on L1.0, 1/ (see Theorem 6.3.6 and the notes to Chapter 6) and we recall that a fully symmetric functional ' 2 M1,1.M, / is normalized if '.A/ D 1 whenever .t, A/ D .1 C t/1, t>0 (atomless case), or .n, A/ D .1 C n/1, n 0(atomic case). It is proved in Theorem 8.2.4 and Theorem 8.6.4 that ! and , when linear, are normalized fully symmetric functionals on M1,1.M, /. It is therefore quite natural to ask whether every normalized fully symmetric functional on M1,1.M, /is of the form ! (or , respectively). Firstly, in Theorem 8.2.4 we prove that if ! is dilation invariant, then the functional ! extends to a normalized fully symmetric functional on M1,1.M, /. Secondly, in Theorem 8.3.6 we show that in fact every normalized fully symmetric functional on M1,1.M, / coincides with some !,where! is dilation invariant. Thus, we can conclude that the set of functionals ! where ! is dilation invariant exactly coincides with the set of all Dixmier traces. Theorem 8.2.9 shows that ! D ! 246 Chapter 8 Heat Kernel Formulas and -function Residues

if ! is an M -invariant extended limit on L1.0, 1/. The theorem also shows that the question as to whether the equality ! D ! holds for every dilation invariant extended limit ! is answered in the negative. Surprisingly, the analogous results fail for -functions. Even though Theorem 8.6.8 conﬁrms that ! D !ılog, for those dilation invariant extended limits ! such that ! ı log is also dilation invariant, Section 8.7 shows that the set of fully symmetric functionals of the form , is an extended limit on L1.0, 1/, is strictly smaller than the set of all normalized fully symmetric functionals. Hence, the -function residues provide a smaller set of traces on M1,1.M, /than either the heat kernel formulas or the Dixmier traces.

8.2 Heat Kernel Functionals

Fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semiﬁnite trace .LetM1,1.M, /be the Dixmier–Macaev operator ideal associated to the pair .M, /,thatis

M M M 1,1. , / :DfA 2 : .A/ 2 M1,1g , kAkM1,1 :Dk.A/kM1,1 , where M1,1 is the Lorentz function space Z 1 t M1,1 :D x 2 L1.0, 1/ : kxkM1,1 :D sup .s, x/ds < 1 , t>0 log.1 C t/ 0 see also Example 2.6.10. Throughout this chapter the symbol M : L1.0, 1/ ! L1.0, 1/ denotes the logarithmic mean Z 1 t ds .M x/.t/ :D x.s/ , x 2 L1.0, 1/, t>0. log.t/ 1 s

Deﬁnition 8.2.1. For every extended limit ! on L1.0, 1/, the functional ! : M1,1.M, /C ! R deﬁned by setting 1 .tA/1 .A/ :D .! ı M/ .e / ,0 A 2 M 1.M, /, ! t 1, is called a heat kernel functional. The following lemma is a variant of the Jensen inequality. We refer the reader to [98] for its proof (see Lemma II.3.4 there).

Lemma 8.2.2. Let F : RC ! RC be a convex function such that F.0/ D 0. If A, B 2 M are positive operators such that B A, then .F.B// .F.A//. In particular, F.A/ 2 L1.M, / implies that F.B/ 2 L1.M, /. Section 8.2 Heat Kernel Functionals 247

The following lemma shows that a heat kernel functional is well deﬁned.

Lemma 8.2.3. If 0 A 2 M1,1.M, /, then 1 .tA/1 M .e / 2 L1.0, 1/. t

For every extended limit ! on L1.0, 1/, we have 1 1 .A/ D ! .Ae.tA/ / . ! log.1 C t/

Proof. We break the proof into a few steps. M M (a) Let 0 A, C 2 1,1. , / be such that .s, C/ DkAkM1,1 =.1 C s/, s>0. 1 The mapping z ! zez , z>0, is convex. Hence, by Lemma 8.2.2, Z 1 1 1 1 .Ae.tA/ / .Ce.tC / / D .s, Ce.tC / /ds. 0 It now follows from Corollary 2.3.17 (d) that Z Z 1 z .tA/1 .t.s,C//1 e dz .Ae / .s, C/e ds DkAkM 1 . 1, k kM 1 z 0 .t A 1,1/

.tA/1 .tA/1 Therefore, Ae 2 L1 and .Ae / D O.log.1 C t//, t>0. (b) By deﬁnition of M ,wehave Z t 1 .tA/1 1 .sA/1 ds M .e / D .e / 2 . t log.t/ 1 s Note that Z Z t 1 .sA/1 ds uA1 .tA/1 e 2 D e du D Ae , t>0. 0 s 1=t Hence, 1 1 1 1 1 M .e.tA/ / D .Ae.tA/ / .AeA / . t log.t/

(c) The right-hand side of the last equality in (b) is a continuous (even differentiable) function of t>0. The continuity at t D 0 is obvious. Thus, it is locally bounded. The ﬁrst assertion of the lemma now follows from (a). The second assertion follows from the ﬁnal display above. 248 Chapter 8 Heat Kernel Formulas and -function Residues

Theorem 8.2.4. For every dilation invariant extended limit ! on L1.0, 1/, ! extends to a fully symmetric linear functional on M1,1.M, /.

Proof. By Lemma 8.2.3, the functional ! is well deﬁned on M1,1.M, /C.We break the proof into steps.

(a) Let 0 A, B 2 M1,1.M, / be such that .B/ D 21=2.A/.Wehave

1 1 .Be.tB/ / D .Ae.2tA/ /, t>0.

It now follows from Lemma 8.2.3 that 1 1 1 1 .B/ D ! .Be.tB/ / D ! .Ae.2tA/ / . ! log.1 C t/ log.1 C 2t/

Since ! is dilation invariant, it follows that 1 1 .B/ D .! ı / .Ae.tA/ / D .A/. ! 1=2 log.1 C t/ !

z1 (b) Let 0 A, B 2 M1,1.M, /, be such that B A. The mapping z ! ze , z>0, is convex. Hence, by Lemma 8.2.2,

1 1 .Be.tB/ / .Ae.tA/ /.

By Lemma 8.2.3, ! .B/ !.A/ for every extended limit !.

A ˚ B A C B 21=2.A ˚ B/.

It now follows from (a) and (b) that

!.A C B/ D ! .A ˚ B/ D !.A/ C ! .B/.

(d) By (c), ! is an additive functional on M1,1.M, /C.Since! is also monotone, it follows that it extends to a linear functional on M1,1.M, /. The latter is fully symmetric by (b).

It appears to be a difficult task to describe the set of all extended limits ! for which Definition 8.2.1 defines a linear functional ! . The following theorem explains why we can restrict consideration to the dilation invariant extended limits. Section 8.2 Heat Kernel Functionals 249

Theorem 8.2.5. If an extended limit ! on L1.0, 1/ is such that ! is a linear functional on M1,1.M, /, then there exists a dilation invariant extended limit !0 such that ! D !0 .

Proof. Deﬁne the mapping S : M1,1.M, /C ! L1.0, 1/ by setting

1 .tA/1 .SA/.t/ :D .Ae /,0 A 2 M 1.M, /C. log.1 C t/ 1,

Let Y be the linear span of the elements SA,where0 A 2 M1,1.M, /,and 0 L1.0, 1/.Wehave 1 0 M M s1 .SA/ 2 s S.sA/ C L1.0, 1/,0 A 2 1,1. , /.

Therefore, Y is a dilation invariant linear subspace of L1.0, 1/. Suppose ! generates a linear functional on M1,1.M, /. In particular, ! .sA/ D s! .A/ for every s>0. It follows from Lemma 8.2.3 that, for 0 A 2 M1,1.M, /,

1 1 !.SA/ D !.A/ D s !.sA/ D s !.S.sA// D .! ı s1 /.SA/.

Thus, !jY is a dilation invariant linear functional. By the deﬁnition of an extended limit, !jY lim sup . Thus, the subspace Y , the dilation semigroup s, s>0, the linear functional !jY and the convex functional lim sup on L1.0, 1/ satisfy the conditions of Theorem 6.2.5. Thus, by Theorem 6.2.5, !jY extends to a dilation invariant extended limit !0 on L1.0, 1/. In particular, ! D !0 .

The next lemma is elementary. We include a detailed computation for completeness.

Lemma 8.2.6. For every positive operator A 2 M and t>0, we have 1 1 1 1 1 A C 3Ae.tA=2/ 4Ae.tA/ A C 3Ae.2tA/ . t C t C

Proof. We claim that

2s s s=2 .1 s/C C 3e 4e .1 s/C C 3e , s 0.

Indeed, if s 1, then the inequality is obvious. Hence, it is sufﬁcient to verify that

3e2s 4es s 1 3es=2 4es s, s 2 Œ0, 1.

Since the left-hand (respectively, right-hand) side of this inequality is convex (respectively, concave) on the interval Œ0, 1 and since the inequality holds for s D 0andfor s D 1, it follows that the inequality remains valid for s 2 Œ0, 1. This proves the claim. 250 Chapter 8 Heat Kernel Formulas and -function Residues

Substituting s D .t/1, we obtain 1 1 1 1 1 C 3e.t=2/ 4e.t/ C 3e.2t/ . t C t C The assertion follows from the Spectral Theorem (see Theorem 2.1.5).

The following lemma delivers a simpler expression for !. It is the crucial component of the proofs in the following sections.

Lemma 8.2.7. For every 0 A 2 M1,1.M, / and for every dilation invariant extended limit ! on L1.0, 1/, we have 1 1 ! .A/ D ! A . (8.8) log.1 C t/ t C

Proof. Denote, for brevity, the right-hand side of (8.8) by !.A/. It follows from Lemma 8.2.6 and Lemma 8.2.3 that 1 3 .A/ C 6 A 4 .A/ .A/ C .2A/. ! ! 2 ! ! 2 ! Since the extended limit ! is dilation invariant, then it follows from Theorem 8.2.4 that !.sA/ D s!.A/ for every s>0 and, therefore, !.A/ D !.A/.

The next lemma concerns the behavior of the distribution function nA of an operator A 2 M1,1.M, /.

Lemma 8.2.8. For every 0 A 2 M1,1.M, / and every dilation invariant extended limit ! on L1.0, 1/, we have 1 1 ! n D 0. t log.1 C t/ A t Proof. It follows from the dilation invariance of ! that 1 1 1 1 ! A D ! A . log.1 C t/ t C log.1 C 2t/ 2t C On the other hand, we have log.1 C 2t/ D .1 C o.1// log.1 C t/ and, therefore, 1 1 1 1 ! A D ! A . log.1 C 2t/ 2t C log.1 C t/ 2t C

Therefore, 1 1 1 ! A A D 0. log.1 C t/ 2t C t C Section 8.2 Heat Kernel Functionals 251

Since 1 1 1 1 1 1 1 1 1 A A D nA C A EA , nA , 2t C t C 2t t t 2t t 2t t the assertion follows immediately.

We are ready to present the last main result of this section. For any dilation invariant extended limit ! on L1.0, 1/, deﬁne a Dixmier trace ! on M1,1.M, / by linear extension of the formula Z 1 t !.A/ D ! .s, A/ds ,0 A 2 M1,1.M, /. log.1 C t/ 0

By the results of Chapter 6, and comments in the end notes of that chapter, ! is a continuous singular trace on M1,1.M, / and, if M is atomless, or atomic, then the set of Dixmier traces !, ! is a dilation invariant extended limit, coincides with the set of all normalized fully symmetric functionals on M1,1.M, /. The next theorem links Dixmier traces on the Dixmier–Macaev operator ideal to heat kernel formulas, and also shows the limitations to their direct connection. This result will be an important part of the proof of Corollary 8.5.2.

Theorem 8.2.9. If ! is a dilation invariant extended limit on L1.0, 1/ such that ! D ! ı M (or ! D ! ı Ps, s>0), then ! D !. There exists a dilation invariant extended limit ! on L1.0, 1/ such that ! ¤ !.

Proof. Let A 2 M1,1.M, / be an arbitrary positive operator. By the spectral theorem, we have Z 1 1 1 1 1 1 A D udnA.u/ C nA . log.1 C t/ t C log.1 C t/ 1=t t log.1 C t/ t

It now follows from Lemma 8.2.7 and Lemma 8.2.8 that Z 1 1 !.A/ D ! udnA.u/ . log.1 C t/ 1=t

The ﬁrst assertion now follows from Theorem 7.4.3. The second assertion follows from Theorem 7.4.7. 252 Chapter 8 Heat Kernel Formulas and -function Residues

8.3 Fully Symmetric Functionals are Heat Kernel Functionals

It follows from Theorem 8.2.4 that the heat kernel functional ! defined in Defini- tion 8.2.1 is a normalized fully symmetric functional on M1,1.M, / whenever ! is a dilation invariant extended limit on L1.0, 1/. In this section, we show the converse. The statement of Lemma 8.2.7 suggests to define a (nonlinear) operator

T : M1,1.M, /C ! L1.0, 1/ by the formula 1 1 .TA/.t/ :D A , t>0. (8.9) log.1 C t/ t C We need some properties of the operator T . Firstly, we show that it is additive on certain pairs of operators 0 A, B 2 M1,1.M, /.

Lemma 8.3.1. For every pair of positive operators A, B 2 M1,1.M, /, we have T.A˚ B/ D TA C TB. Proof. Immediately from the assumption we have for t>0that 1 1 1 A ˚ B D A C B . t C t C t C The result follows due to the linearity of .

Corollary 8.3.2. The set Y D T M1,1.M, /C T M1,1.M, /C is a dilation invariant (not necessarily closed) linear subspace of L1.0, 1/.

Proof. If z1, z2 2 Y , then there exist positive operators A, B, C , D 2 M1,1.M, / such that z1 D TA TB and z2 D TC TD. By Lemma 8.3.1, we have

z1 C z2 D TA C TC TB TD D T.A˚ C/ T.B ˚ D/ 2 Y .

For every >0, we have that .A/, .B/ are positive elements from M1,1 and therefore

z1 D TA TB D T..A// T..B// 2 Y .

Similarly, for every <0, we have

z1 DjjTB jjTA D T.jj.B// T.jj.A// 2 Y .

Thus, Y is a linear subspace of L1.0, 1/. Section 8.3 Fully Symmetric Functionals are Heat Kernel Functionals 253

It follows from the deﬁnition (8.9) that for every positive A 2 M1,1.M, / and for every t, s>0, we have t 1 s s 1 .TA/ D A D s1A . s log.1 C t=s/ t C log.1 C t=s/ t C

Therefore, 1 0 s.TA/ 2 sT.s A/ C L1.0, 1/.

Hence, the subspace Y is dilation invariant.

Next, we explain the connection of the operator T with fully symmetric functionals on M1,1.M, /. We continue to use the notation introduced in Corollary 8.3.2.

Lemma 8.3.3. For every symmetric functional ' on M1,1.M, /, there exists a unique linear functional : Y ! R such that ' D ı T on M1,1.M, /C.

Proof. If z 2 Y is such that z D TA TB for 0 A, B 2 M1,1.M, /,thenset

.z/ :D '.A/ '.B/.

First, we prove that is well deﬁned. Let z 2 Y be such that z D TA TB and z D TC TD for 0 A, B, C , D 2 M1,1.M, /. It follows from the deﬁnition of T and Lemma 8.3.1 that

T.A˚ D/ D TA C TD D TB C TC D T.B ˚ C/.

Thus, A ˚ D is equimeasurable with B ˚ C . It follows that

'.A/ C '.D/ D '.A ˚ D/ D '.B ˚ C/D '.B/ C '.C/.

Hence, '.A/ '.B/ D '.C/ '.D/.

This proves that is a well-deﬁned functional. Next, we prove the linearity of .Letz1 D TA TB and let z2 D TC TD.It follows from the deﬁnition of T and Lemma 8.3.1 that

z1 C z2 D TA C TC TB TD D T.A˚ C/ T.B ˚ D/.

Therefore,

.z1Cz2/ D '.A˚C/'.B˚D/ D .'.A/'.B//C.'.C /'.D// D .z1/C.z2/. 254 Chapter 8 Heat Kernel Formulas and -function Residues

If 0, then

.z1/ D .T ..A// T..B/// D '..A// '..B// D .'.A/ '.B// D .z1/.

Similarly, if 0, then

.z1/ D .T .jj.B// T.jj.A/// D '.jj.B// '.jj.A//

Djj.'.B/ '.A// D .z1/.

Hence, the functional : Y ! R is linear.

Lemma 8.3.4. Let 0 A, B 2 M1,1.M, / be such that TB TA. For every fully symmetric functional ' on M1,1.M, /, we have '.B/ '.A/.

Proof. It follows immediately from the deﬁnition (8.9) that the assumption TB TA is equivalent to 1 1 B A , 8t>0. t C t C

Applying Theorem 3.3.6 we obtain B A and, therefore, '.B/ '.A/.

Lemma 8.3.5. Let 0 A, B 2 M1,1.M, /. For every normalized fully symmetric functional ' on M1,1.M, /, we have

'.B/ '.A/ lim sup.TB TA/.t/. (8.10) t!1

1 Proof. Let 0 C 2 M1,1.M, / be an operator such that .t, C/ D .1 C t/ , t>0. Let c be an arbitrary real number greater than the right-hand side of (8.10). It follows from the deﬁnition of T that

0 .T .jcjC //.t/ 2jcjCL1.0, 1/.

We split the proof into the cases c 0andc<0. If c 0, then it follows from Lemma 8.3.1 that

lim sup.TB T.A˚ cC //.t/ Dc C lim sup.TB TA/.t/ < 0. t!1 t!1

Hence, there exists N>0 such that we have

.TB/.t/ .T .A ˚ cC //.t/, 8t>N. Section 8.3 Fully Symmetric Functionals are Heat Kernel Functionals 255

It now follows from the deﬁnition of T : M1,1.M, /C ! L1.0, 1/ that 1 T min B, T.A˚ cC/. N

Since ' is singular and fully symmetric, it follows from Lemma 8.3.4 that 1 '.B/ D ' min B, '.A ˚ cC/ D c C '.A/. N

Since c is an arbitrary number exceeding the right-hand side, the inequality (8.10) follows. If c<0, then it follows from Lemma 8.3.1 that

lim sup.T .B ˚cC/ TA/.t/ Dc C lim sup.TB TA/.t/ < 0. t!1 t!1

Hence, there exists N>0 such that we have

.T .B ˚cC //.t/ .TA/.t/, 8t>N.

It now follows from the deﬁnition of T : M1,1.M, /C ! L1.0, 1/ that 1 T min B ˚cC, T.A/. N

By Lemma 8.3.4, we have 1 '.B/ D c C '.B ˚cC/ D c C ' min B ˚cC, c C '.A/. N

Since c is an arbitrary number exceeding the right-hand side, the inequality (8.10) follows.

The following theorem is the main result of this section. It shows that an arbitrary fully symmetric functional ' on M1,1.M, / can be represented (up to a scalar) as a heat kernel formula.

Theorem 8.3.6. Let ' be a normalized fully symmetric functional on M1,1.M, /. There exists a dilation invariant extended limit ! on L1.0, 1/ such that ' D !.

Proof. Let Y be the linear subspace of L1.0, 1/ deﬁned in Corollary 8.3.2 and let : Y ! R be the linear functional constructed in Lemma 8.3.3. By Lemma 8.3.5, we have .z/ lim sup.z/ for every z 2 Y . By the Hahn–Banach theorem, is an extended 256 Chapter 8 Heat Kernel Formulas and -function Residues

limit on L1.0, 1/ (restricted to the subspace Y ). Arguing as in Corollary 8.3.2, for s>0 we obtain that

1 0 s.TA/ 2 sT.s A/ C L1.0, 1/.

0 Since the extended limit vanishes on L1.0, 1/, it follows that

1 1 .s.TA// D s.T .s A// D s'.s A/ where the last equality uses Lemma 8.3.3. Since ' is a fully symmetric functional then

s'.s1A/ D '.A/ D .TA/ where the last equality uses Lemma 8.3.3 again. Thus, .sz/ D .z/,forallz 2 Y . By Theorem 6.2.5, there exists a dilation invariant extended limit ! on the space L1.0, 1/ such that ! D on Y . In particular, !.TA/ D .TA/ D '.A/,forall 0 A 2 M1,1.M, /. By the deﬁnition of T ,wehave 1 1 '.A/ D ! A , 80 A 2 M1,1.M, /. log.1 C t/ t C

The assertion now follows from Lemma 8.2.7.

8.4 Generalized Heat Kernel Functionals

Let ! be an extended limit on L1.0, 1/ and let B 2 M. Following [9], we consider the functionals on M1,1.M, /C deﬁned by the formula 1 .A/ :D .! ı M/ t ! .f.tA/B/ , A 0 (8.11) !,B,f t where f is a function on the semi-axis. This section shows the validity of the deﬁnition (8.11) for a twice-differentiable bounded function f such that f.0/ D f 0.0/ D 0.

Theorem 8.4.1. Let f 2 C 2Œ0, 1/ be a bounded function such that f.0/ D f 0.0/ D 0.Let0 A 2 M1,1.M, / and let B 2 M. We have 1 M t ! .f.tA/B/ 2 L1.0, 1/. t

To prove the theorem we require the following lemmas. Section 8.4 Generalized Heat Kernel Functionals 257

Lemma 8.4.2. If 0 A 2 M 1.M, /, then we have 1, 1 (a) A t C D O.log.t// as t !1. ˚ 1 2 1 (b) min A, t D O t log.t/ as t !1.

Proof.Letc :DkAkM1,1 and set z.s/ D 1=.1 C s/, s>0. We have A cz. (a) It follows from Theorem 3.3.6 that 1 1 A cz D O.log.t// t C t C as t !1.

(b) For ﬁxed t>0, we have 1 2 1 1 X 1 1 min A, D n C A2E , t t 2 A t A 2nC1t 2nt n0 1 1 X 1 1 1 n C E , t 2 A t 22nt 2 A 2nC1t 2nt n0 1 1 X 1 1 n C n . t 2 A t 22nt 2 A 2nC1t n0

We have .s, A/ c log.1 C s/=s for every s>0. Therefore, 1 c log.1 C s/ 1 n m s : 2culog.u/ A u s u

for all sufﬁciently large u. It follows that 1 2 2c X log.t/ min A, .t log.t/ C 22n log.2nC1t// D O . t t 2 t n0

2 Lemma 8.4.3. If f0.t/ :D minf1, t g, t>0, then 1 t ! M .f .tA// 2 L1.0, 1/,0 A 2 M 1.M, /. t 0 1,

Proof. For a ﬁxed t>0, we have Z Z t t 1 1 2 2 1 2 2 M .f0.tA// D .minfA , s g/ds D minfA , s gds . t log.t/ 1 log.t/ 1 258 Chapter 8 Heat Kernel Formulas and -function Residues

If >t1,then Z Z Z t 1 t 2 2 2 ds 1 minf , s gds D ds C 2 D 2 . 0 0 1 s t

If t 1,then Z Z t t minf2, s2gds D 2ds D 2t. 0 0 In either case, we have Z t 1 1 2 minf2, s2gds D 2 C t min , . 0 t C t

It follows from the Spectral Theorem that Z t 1 1 2 minfA2, s2gds D 2 A C t min A, . 0 t C t

In particular, 1 2 1 t 1 2 1 M .f0.tA// D A C min A, CO . t log.t/ t C log.t/ t log.t/

The assertion follows from Lemma 8.4.2.

We can now prove Theorem 8.4.1.

2 Proof of Theorem 8.4.1. Set f0.t/ :D minf1, t g, t>0. Observe that the assumptions on f guarantee that there exists a constant c>0suchthatjf.t/jcf0.t/. It is now immediate that

j.f.tA/B/jkBk1.jf.tA/j/ ckBk1.f0.tA//.

The assertion follows from Lemma 8.4.3.

8.5 Reduction of Generalized Heat Kernel Functionals

The last section deﬁned the generalized heat kernel functional for a dilation invariant extended limit ! on L1.0, 1/, an arbitrary operator B 2 M, and a bounded function f 2 C 2Œ0, 1/ such that f.0/ D f 0.0/ D 0, 1 .A/ :D .! ı M/ t ! .f.tA/B/ ,0 A 2 M 1.M, /. !,B,f t 1, Section 8.5 Reduction of Generalized Heat Kernel Functionals 259

This section shows that this general heat kernel functional can be reduced to a scalar multiple of the weighted heat kernel functional !.B/ from Section 8.2.

Theorem 8.5.1. Let f 2 C 2Œ0, 1/ be a bounded function such that f.0/ D f 0.0/ D 0. Let 0 A 2 M1,1.M, / and let B 2 M. For every dilation invariant extended limit ! on L1.0, 1/ we have Z 1 ds !,B,f .A/ D f.s/ 2 ! .AB/. (8.12) 0 s The proof of the theorem is given below. The following corollary treats the case of classical heat kernel formulas and their connection to Dixmier traces.

Corollary 8.5.2. Let 0 A 2 M1,1.M, / and let B 2 M. For every dilation invariant extended limit ! on L1.0, 1/ we have 1 1 .! ı M/ .exp..tA/q/B/ D 1 C .AB/. t q !

If, in addition, ! D ! ı M , then 1 1 .! ı M/ .exp..tA/q/B/ D 1 C .AB/. t q !

Here is the Gamma function and q>0.

Proof. Use f : s ! exp.sq /, s>0, in Theorem 8.5.1 and observe that Z 1 ds 1 f.s/ D 1 C . 2 0 s q

The ﬁrst assertion follows from Theorem 8.5.1. Combining this with Theo- rem 8.2.9, we obtain the second assertion.

The following lemmas are used to prove Theorem 8.5.1.

Lemma 8.5.3. If 0 A 2 M1,1.M, / and B 2 M, then 1 1 ! A B D ! .AB/ (8.13) log.1 C t/ t C for every extended limit !. 260 Chapter 8 Heat Kernel Formulas and -function Residues

Proof. Recall that the function u ! .u 1=t/C is convex. It follows from Theo- rem A.2.7 in the appendix that 1 1=2 1=2 1 (a) A t CB B AB t C if 0 B 1. 1 1=2 1=2 1 (b) A t CB B AB t C if B 1. Hence, we have 1 1 1 1 ! A B ! B1=2AB1=2 log.1 C t/ t C log.1 C t/ t C (8.14) for 0 B 1and 1 1 1 1 ! A B ! B1=2AB1=2 log.1 C t/ t C log.1 C t/ t C (8.15) for B 1. Since both sides are homogeneous, the inequality (8.14) is valid if B 0, while the inequality (8.15) is valid if B is bounded from below by a strictly positive constant. Thus, we have the equality (8.13) valid for every B bounded from below by a strictly positive constant. Set Bn :D maxfB,1=ng, n 1. It follows that equality (8.13) holds with B replaced with Bn throughout. Clearly, ˇ ˇ ˇ ˇ ˇ 1 1 1 1 ˇ 1 ˇ! A Bn ! A B ˇ ! .A/. log.1 C t/ t C log.1 C t/ t C n

Therefore, 1 1 1 1 ! A B D lim ! A Bn !1 log.1 C t/ t C n log.1 C t/ t C

D lim ! .ABn/ D !.AB/. n!1

Lemma 8.5.4. If 0 A 2 M1,1.M, / and B 2 M, then 1 s .! ı M/ E , 1 B D s1 .AB/ t A t ! for every dilation invariant extended limit ! and for every s>0.

Proof. We have Z 1 s 1 t s du M EA , 1 B D EA , 1 2 B . t t log.t/ 1 u u Section 8.5 Reduction of Generalized Heat Kernel Functionals 261

However, Z t s du 1 1 EA , 1 2 D s A . 0 u u t C Therefore, 1 s 1 1 1 .! ı M/ EA , 1 B D ! s A B . t t log.1 C t/ t C Since ! is dilation invariant, the assertion follows from Lemma 8.5.3.

Lemma 8.5.5. Let f : RC ! R be supported and monotone on .a, b. Let 0 A 2 M1,1.M, / and let B 2 M. For every dilation invariant extended limit ! on L1.0, 1/ we have Z b ds .A/ D f.s/ .AB/. !,B,f 2 ! a s Proof. Without loss of generality, we may assume that f is increasing on .a, b and that B 0. Let a D a0 a1 a2 an D b.Sincef is increasing on .a, b,wehave for every t>0 Xn1 Xn1 ak akC1 ak akC1 f.a /E , f.tA/ f.a C /E , . k A t t k 1 A t t kD0 kD0 Therefore, Xn1 1 ak akC1 .A/ f.a C /.! ı M/ E , B !,B,f k 1 t A t t kD0 and Xn1 1 a a C .A/ f.a /.! ı M/ E k , k 1 B . !,B,f k t A t t kD0

We have a a C a a C E k , k 1 D E k , 1 E k 1 , 1 . A t t A t A t

It follows from Lemma 8.5.4 that 1 ak akC1 1 1 .! ı M/ EA , B D ! .AB/. t t t ak akC1 262 Chapter 8 Heat Kernel Formulas and -function Residues

Hence,

1 Xn 1 1 f.ak/ ! .AB/ !,B,f .A/ ak akC1 kD0 Xn 1 1 1 f.akC1/ !.AB/. ak akC1 kD0 R b 2 Both coefﬁcients in the latter formula converge to a f.s/s ds. The proof of Theorem 8.5.1 can now be given.

Proof of Theorem 8.5.1. Without loss of generality, f 0. For a given n 2 N,select 2 fn 2 C Œ0, 1/ supported on .1=2n,2n such that 0 fn f andsuchthatf D fn on .1=n, n. By assumption, jf.t/jconst minf1, t 2g, t>0. It follows that 0 f fn const gn,where

2 2 2 2 gn.t/ :D minft , n gCminf1, t n g, t>0.

We have

0 !,B,f .A/ !,B,fn .A/ const kBk1!,1,gn .A/.

Since ! is a dilation invariant extended limit, it follows that 1 .A/ D z.! ı M/ .minf.tA/2, n2g/ !,1,gn t 1 C .! ı M/ .minf1, n2.tA/2g/ t 1 1 D .! ı M ı / min .tA/2, n t n2 1 1 C .! ı M ı / min 1, .tA/2 1=n t n2 2 1 1 D .! ı M/ .minf1, t 2g/ D O . n t n

Hence, 1 0 .A/ .A/ O . !,B,f !,B,fn n

Therefore, !,B,fn .A/ ! !,B,f .A/ as n !1. Section 8.6 -function Residues 263

Every function from C 2Œ1=2n,2n is of bounded variation on Œ1=2n,2n. In particular, fnj.1=2n,2n is a difference of two monotone (on the interval .1=2n,2n) functions. By Lemma 8.5.5, the equality (8.12) holds for fn. Hence, it also holds for the function f .

8.6 -function Residues

We now shift our attention to -function residues, which are the functionals ! on M1,1.M, /, ! an extended limit on L1.0, 1/, introduced in the following definition. Observe that, when considering -function residues, we do not restrict our attention only to ! that are dilation invariant. We show in this section that -function residues provide fully symmetric functionals on M1,1.M, / and can be identiﬁed with Dixmier traces for certain !.

Deﬁnition 8.6.1. For every extended limit ! on L1.0, 1/, the functionals ! : M1,1.M, /C ! RC and !,B : M1,1.M, /C ! C are deﬁned by the formulas 1 1 .A/ :D ! .A1C1=t/ , .A/ :D ! .A1C1=tB/ ! t !,B t

0 A 2 M1,1.M, /.

Here, B 2 M is an arbitrary operator. We begin by showing that the functionals given in Deﬁnition 8.6.1 are well deﬁned on M1,1.M, /C.

Lemma 8.6.2. If ! : L1.0, 1/ ! R is an extended limit, then !.A/ < 1 and !,B.A/ < 1 for any 0 A 2 M1,1.M, /.

1 Proof. It is clear that .s, A/ .1 C s/ kAkM1,1 , s>0. Therefore, Z 1 1C1=t 1C1=t dt 1C1=t .A / kAkM D tkAkM . 1,1 1C1=t 1,1 0 .1 C s/

Hence, !.A/ kAkM1,1 . It follows from

1C1=t 1C1=t j.A B/jkBk1.A / that !,B.A/ kBk1!.A/.

Lemma 8.6.3. For any 0 A, C 2 M1,1.M, / we have

.A1Cs C C 1Cs/ ..A C C/1Cs/ 2s.A1Cs C C 1Cs/, s>0. 264 Chapter 8 Heat Kernel Formulas and -function Residues

Proof. In the special case when M D L.H /, the ﬁrst inequality can be found in [138, (2.9)]. In the general case, it follows directly from Proposition 4.6 (ii) of [92] when f.u/ D u1Cs, u>0. The second inequality follows from the same proposition by setting there a D a D b D b D 21=2.

The following theorem shows that the functionals ! are fully symmetric on M1,1.M, /.

Theorem 8.6.4. If ! : L1.0, 1/ ! R is an extended limit, then ! extends to a fully symmetric linear functional on M1,1.M, /.

Proof. It follows from the left-hand side inequality of Lemma 8.6.3 that, for all positive operators A, C 2 M1,1.M, /,wehave

!.A C C/ !.A/ C !.C /.

Noting that !.j21=t 1j/ D 0, it follows from the right-hand side inequality of Lemma 8.6.3 and Remark 6.3.5 that

!.A C C/ !.A/ C !.C /.

Therefore, we have

!.A C C/D !.A/ C !.C /.

In other words, the functional ! is additive on M1,1.M, /C. The homogeneity of ! follows from Remark 6.3.5. Thus, ! admits a unique extension to a linear functional on M1,1.M, /. Finally, if C 0andC A 2 M1,1.M, /C,thenC , A 2 L1Cs.M, / and 1Cs 1Cs 1 1C1=t 1 1C1=t .C / .A /. Hence, t .C / t .A / and so !.C / !.A/. Hence, ! is a fully symmetric linear functional.

The following is our main result concerning the -function residue.

Theorem 8.6.5. If ! : L1.0, 1/ ! R is an extended limit, then

!,B.A/ D !.AB/,0 A 2 M1,1.M, /, B 2 M.

Proof. Suppose ﬁrst that m B M for some strictly positive constants m, M . Applying Lemma A.2.9 in the appendix to the operators A and M 1B (respectively, m1B), we have

msB1=2A1CsB1=2 .B1=2AB1=2/1Cs M sB1=2A1CsB1=2. Section 8.6 -function Residues 265

Therefore, 1 1 1 m1=t.A1C1=tB/ ..B1=2AB1=2/1C1=t/ M 1=t.A1C1=tB/. t t t Since !.jm1=t 1j/ D 0and!.jM 1=t 1j/ D 0, it follows from Remark 6.3.5 that 1=2 1=2 !,B.A/ D !.B AB /. By Theorem 8.6.4, ! is a symmetric functional and, 1=2 1=2 therefore, by Lemma 2.7.4 is a trace. It follows that !.B AB / D !.AB/.This proves the assertion of the theorem when B is bounded above and below by strictly positive constants. Let now 0 B 2 M be arbitrary and set Bn :D maxfB,1=ng, n 1. From the ﬁrst part of the proof, we have

!,Bn .A/ D !.ABn/.

Since ABn ! AB in M1,1.M, / and since ! is a bounded functional (by Theo- rem 8.6.4), it follows that !.ABn/ ! !.AB/ as n !1. On the other hand,

j!,Bn .A/ !,B.A/jDj!,BnB .A/jkBn Bk1!.A/ ! 0, n !1. Therefore

!,B.A/ D lim !,Bn .A/ D lim !.ABn/ D !.AB/. n!1 n!1 The assertion for arbitrary B 2 M follows by linearity.

We now provide several formulas linking Dixmier traces and -function residues.

Lemma 8.6.6. Let f : .0, 1/ ! R (respectively, ˇ : .0, 1/ ! R) be a positive decreasing (respectively, increasing) function. If ˇ.t/ t for all t>0, then Z Z 1 1 f.t/dˇ.t/ f.t/dt. 0 0

Proof. Without loss of generality, ˇ.0/ D 0. Let tk,0 k n, n 1, be an increasing sequence such that t0 D 0. We have

Xn1 Xn1 f.tk/.ˇ.tkC1/ ˇ.tk// D ˇ.tkC1/.f .tk1/ f.tk// C ˇ.tn/f .tn1/ kD0 kD1 Xn1 tkC1.f .tk1/ f.tk// C tnf.tn1/ kD1 Xn1 D f.tk/.tkC1 tk/. kD0 266 Chapter 8 Heat Kernel Formulas and -function Residues

Hence, the Riemann–Stieltjes sums for the left-hand side do not exceed the Riemann– Stieltjes sum for the right-hand side.

Theorem 8.6.7. Let ˇ : RC ! RC be an increasing function such that 0 ˇ.t/ t, t 0. We have Z h.t/ ˇ.t/ 1 ! D ! , where h.t/ :D eu=tdˇ.u/, t>0 (8.16) t t 0 for any dilation invariant extended limit ! on L1.0, 1/.

Proof. First, note that Z Z 1 1 0 eu=t dˇ.u/ eu=tdu D t 0 0 by Lemma 8.6.6. Hence, the left-hand side of (8.16) is well deﬁned. Since ! is dilation invariant, it follows that Z Z 1 1 1 h.t/ 1 h.t/ ! .eu=t/n eu=tdˇ.u/ D ! D snds ! t 0 n C 1 t 0 t for every n 0. Therefore, Z Z 1 1 1 h.t/ ! p.eu=t/eu=t dˇ.u/ D p.s/ds ! t 0 0 t for every polynomial p. For every f 2 CŒ0, 1, there exists a sequence pk, k 0, of polynomials such that pn ! f uniformly on Œ0, 1. Therefore, Z Z 1 1 1 h.t/ ! f.eu=t /eu=tdˇ.u/ D f.s/ds ! (8.17) t 0 0 t for every f 2 CŒ0, 1. 1 Set g.s/ :D s Œe1,1.s/, s 2 Œ0, 1.Wehave Z 1 g.eu=t/eu=tdˇ.u/ D ˇ.t/ ˇ.0/ 0 and, therefore, Z 1 1 ˇ.t/ ! g.eu=t /eu=tdˇ.u/ D ! . (8.18) t 0 t Section 8.6 -function Residues 267

Choose 0 f1k, f2k 2 CŒ0, 1 such that f1k # g and f2k " g almost everywhere. It follows from (8.17) and (8.18) that Z Z 1 h.t/ ˇ.t/ 1 h.t/ f2k.s/ds ! ! f1k.s/ds ! . 0 t t 0 t

Letting k !1, we obtain the assertion.

With the above results on -function residues, we can connect -function residue functionals on M1,1.M, / with Dixmier traces on M1,1.M, /.

Theorem 8.6.8. If ! is a dilation invariant extended limit ! on L1.0, 1/ such that the extended limit ! ı log is still dilation invariant, then ! D !ılog.

Proof. It is sufﬁcient to verify the equality ! D !ılog on positive operators A 2 M1,1.M, / such that A 1. Deﬁne a continuously increasing function ˇ : .0, 1/ ! .0, 1/ by setting Z 1 u ˇ.u/ :D dnA./ D .AEA.e , 1//. eu

u u Since nA.e / D O.ue / as u !1, it follows that

Z u Z u nA.e / cue ˇ.u/ .s, A/ds .s, A/ds D O.u/. 0 0 Let h be as in Theorem 8.6.7 as applied to the above ˇ.Wehave Z Z 1 1 u=t u.1C1=t/ u 1C1=t h.t/ D e dˇ.u/ D e udnA.e / D .A /. 0 0 Since ! ı log is dilation invariant, it follows from Deﬁnition 8.6.1 and Theorem 8.6.7 that h.t/ ˇ.t/ ı .A/ D .! ı log/ D .! ı log/ . (8.19) ! log t t

By Theorem 7.4.3, we have Z 1 1 ˇ.t/ !.A/ D ! dnA./ D .! ı log/ . (8.20) log.t/ 1=t t

The assertion follows immediately from (8.19) and (8.20)

The next corollary follows by combining Theorems 8.6.5 and 8.6.8, we omit the proof. 268 Chapter 8 Heat Kernel Formulas and -function Residues

Corollary 8.6.9. If ! is a dilation invariant extended limit on L1.0, 1/ such that the extended limit ! ı log is still dilation invariant, then 1 1C1=t .AB/ D .! ı log/ .A B/ ,0 A 2 M 1.M, /, B 2 M. ! t 1,

8.7 Not Every Dixmier Trace is a -function Residue

In this section we demonstrate that, unlike the situation with heat kernel functionals (see Theorem 8.3.6), the set of all -function residues is strictly smaller than the set of all normalized fully symmetric functionals. Hence, unlike heat kernels functionals, -function residues do not represent all the Dixmier traces on M1,1.M, /.

Theorem 8.7.1. The set of Dixmier traces on M1,1.M, / is strictly larger than the set of -function residues on M1,1.M, /. The theorem is proved below, ﬁrst we require two lemmas.

Lemma 8.7.2. For the function x 2 CŒ0, 1, deﬁned by the formula X nCu x.u/ :D 2nCu2 , u 2 Œ0, 1, n2Z we have kxk1 < 2= log.2/.

Proof. Evidently, X X nCu u sup x.u/ sup 2nCu2 D sup 2u2 . u2Œ0,1 u2Œ0,1 u2Œn,nC1 n2Z n2Z

A direct computation yields 8 ˆ n2n <ˆ2 , n 1 u sup 2u2 D 1=.e log.2//, n D 0 2 , C1 ˆ u Œn n :ˆ nC1 2nC12 , n<0.

Hence, X 1 n sup x.u/ C 2n2 . (8.21) u2Œ0,1 e log.2/ n2Z

We have X X n 2n2 23=2 C 2n D 23=2 C 21 n<0 n2 Section 8.7 Not Every Dixmier Trace is a -function Residue 269 and X n 2n2 D 201 C 212 C 224 C 238 C 1 C 22 C 24. n0

Hence, 1 29 2 sup x.u/ C C 23=2 < . u2Œ0,1 e log.2/ 16 log.2/

Lemma 8.7.3. For a positive operator A 2 M1,1.M, / such that

k2k k .A/ D sup 2 2 k0 .0,2 / and for x 2 CŒ0, 1 as in Lemma 8.7.2, we have

lim sup s.A1Cs/ D sup x.u/. s#0 u2Œ0,1

Proof. For s>0, we have X k k k1 .A1Cs/ D 2s C 2.k2 /.1Cs/ .22 22 / X k 1 X k k D 2k.1Cs/s2 C O.1/ D 2k.1Cs/s2 C O.1/. k1 k2Z

Thus, X k lim sup s.A1Cs / D lim sup s 2k.1Cs/s2 . s#0 s#0 k2Z

For every s 2 .0, 1/,letm DŒlog2.s/ and u Dflog2.s/g (the negative integer N and fractional parts of log2.s/, respectively). Clearly, m 2 and u 2 Œ0, 1/ are such that s D 2um.Fork D m C n,wehave

k.1 C s/ s2k D2nCu C n.1 C s/ C m.1 C s/.

Hence, for k D m C n,

C k C C um nCu s2k.1 s/ s2 D 2 sŒlog2.s/ 2u n.1 2 / 2 .

Since, s log.s/ ! 0ass ! 0, it follows that X um nCu lim sup s.A1Cs/ D lim sup sup 2uCn.1C2 /2 . (8.22) s#0 m!1 u2Œ0,1 n2Z 270 Chapter 8 Heat Kernel Formulas and -function Residues

We claim the following uniform convergence X X um nCu nCu 2uCn.1C2 /2 2uCn2 , m !1. n2Z n2Z

Indeed, since

um nCu n 2uCn.1C2 /2 21C2n2 , n 0,

um nCu 2uCn.1C2 /2 21Cn, n<0 P uCn.1C2um/2nCu we know that the tails of the series n2Z 2 are uniformly small. Since the individual terms in the sum converge uniformly, the claim is shown. The assertion of the lemma follows from the above claim.

We can now prove Theorem 8.7.1. For convenience we use a simplifying result from the next chapter in the proof.

Proof of Theorem 8.7.1. Let A 2 M1,1.M, / be the operator deﬁned in Lem- ma 8.7.3. It follows from Lemma 8.7.2 and Lemma 8.7.3 that

1Cs 2 sup !.A/ D lim sup s.A /< . ! s#0 log.2/

Set Z t .t/ :D .s, A/ds, t>0. 0 in Lemma 9.2.3. By that lemma, we have that Z Z 1 t 1 t .s, A/ds D lim sup .s, A/ds. log.1 C t/ 0 t!1 log.1 C t/ 0

Here, : L1.0, 1/ ! R is the convex homogeneous functional given by the formula (6.3). An easy computation (taking t D 2n, n !1) shows that Z 1 t 2 lim sup .s, A/ds D . t!1 log.1 C t/ 0 log.2/

By the Hahn–Banach theorem, there exist functionals !0 2 L1.0, 1/ such that !0 and such that Z Z 1 t 1 t 2 !0 .s, A/ds D .s, A/ds D . log.1 C t/ 0 log.1 C t/ 0 log.2/ Section 8.8 Notes 271

By Lemma 6.2.7, !0 is a dilation invariant extended limit and !0 .A/ D 2= log.2/.

Hence, !0 .A/ > sup! ! .A/ and the Dixmier trace !0 is not the residue of any -function.

8.8 Notes

-function Residues and Heat Kernel Functionals The interplay between Dixmier traces ! and the functionals and ! on M1,1.M, / has been an important chapter in noncommutative geometry and has been treated (amongst many papers) in [9,28,32–34,48,212,215]. We look at the history of some of these results. In [32], the equality 1 1C1=t .AB/ D .! ı log/ .A B/ D ılog, .A/,0 A 2 M1,1.M, / (8.23) ! t ! B was established for every B 2 M under strict conditions on !. These conditions were dilation invariance for both ! and !ılog and M -invariance of !. In [33], for the special case B D 1, the assumption that ! is M -invariant was removed. However, the case of an arbitrary B appears to be inaccessible by the methods in that article. In Section 8.6, we proved the general result which implies, in particular, that the equality (8.23) holds without requiring M -invariance of !. Our treatment followed the approach of [241]. In [32], the equality 1 1 ! .exp..tA/q/B/ D 1 C .AB/ (8.24) t q ! was established under the same strict conditions on ! and ! ı log as above. In [215], in the special case B D 1 the equality (8.24) was established under the assumption that ! is M - invariant. However, again the case of an arbitrary B appears to be inaccessible by the methods in that article. In Section 8.5, we treated the case of an arbitrary operator B (again, mostly following [241]). Theorem 8.5.1 extends the results of [32,33] and gives an affirmative answer to the question stated in [9]. Previous articles [32, 33, 215, 241] stating similar results to the combination of Theo- rems 8.2.9 and 8.5.1 were based on the weak Karamata theorem (that is, Theorem 8.6.7) with the implicit condition ˇ.u/ u, u 0. This condition was not explicitly checked in [32, p. 92], [33, p. 276], [215, p. 570], including [241, p. 2466] where it can be seen that the condition nA.1=u/ D O.u/ (which is required to apply the weak Karamata theorem) in fact implies that the operator A 2 L1,1.M, /. That is, in [215, 241] there is an error even though an alternative proof is given in [241, Theorem 49]. All these drawbacks are rectified in Theorem 8.5.1 and Corollary 8.5.2. Corollary 8.6.9 strengthens and extends the results of [33, Theorem 4.11] and [32, Theo- rem 3.8]. Various formulas of noncommutative geometry (in particular, those involving heat kernel estimates and generalized-functions) were established in [32,33,48] when the extended limit ! wasassumedtobeM -invariant. This class of extended limits was first introduced in [32] (see also [66]) and further studied and used in [6,9,33]. Chapter 9 Measurability in Lorentz Ideals

9.1 Introduction

As we did in Chapter 8, we fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace . We explain the results of this chapter using the Dixmier–Macaev operator ideal M1,1.M, /.Thetwomainsetsof Dixmier traces which we have considered in Chapter 7 and Chapter 8 are the set of all Dixmier traces ! (recall from Chapter 6 that this set coincides with the set of all normalized fully symmetric functionals on M1,1.M, /) and the subset of Dixmier traces given by M -invariant extended limits, that is ! where ! D ! ı M is an extended limit on L1.0, 1/. This subdivision in our consideration naturally leads to the following definition.

Deﬁnition 9.1.1. An operator A 2 M1,1.M, / is said to be

(a) Dixmier measurable if the value !.A/ is independent of the choice of the dilation invariant extended limit ! on L1.0, 1/.

(b) M -measurable if the value !.A/ is independent of the choice of the M -invariant extended limit ! D ! ı M on L1.0, 1/.

The objective of this chapter is the characterization of the set of all positive Dixmier measurable operators and the set of all positive M -measurable operators in M1,1.M, /. The main results are Theorems 9.2.1 and 9.3.1 and Theorem 9.5.1. The- orem 9.2.1 asserts that a positive operator A 2 M1,1.M, / is Dixmier measurable if and only if there exists the limit Z 1 t lim .s, A/ds. (9.1) t!1 log.1 C t/ 0

For a Dixmier measurable operator A, this limit calculates the value of every Dixmier trace of A. In Theorem 9.3.1, an equivalent condition is stated in terms of -function residues and heat kernel functionals. Theorem 9.5.1 provides the description of the positive M -measurable operators. More precisely, an operator 0 A 2 M1,1.M, / is M -measurable if and only if Z 1 t M n .s, A/ds ! const, n !1 log.1 C t/ 0 Section 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals 273

0 in the Calkin algebra L1.0, 1/=L1.0, 1/. Using these descriptions Theorem 9.5.4 asserts that the two sets introduced in Deﬁnition 9.1.1 are distinct, that is, there exists 0 A 2 M1,1.M, / which is M -measurable but not Dixmier-measurable. Knowing the Lidskii formula for a Dixmier trace (see Theorem 7.3.1) and the measurability result (9.1), it is natural to ask whether an arbitrary compact operator A in the Lorentz ideal M1,1 :D M1,1.L.H /,Tr/ of compact operators is Dixmier measurable if and only if the limit

1 Xn lim .j , A/ n!1 log.2 C n/ j D0 exists, where .A/ is an eigenvalue sequence of A. Section 9.7 shows that this proposition is true if A 2 L1,1 where L1,1 :D L1,1.L.H /,Tr/ is the weak-l1 ideal of compact operators. It also provides a counterexample that shows the result is not true for an arbitrary operator A 2 M1,1. The notion of Dixmier measurability plays an important role in Part IV, and its use will be observed in subsequent chapters.

9.2 Positive Dixmier Measurable Operators in Lorentz Ideals

For this chapter, ﬁx an atomless or atomic von Neumann algebra M equipped with a faithful normal semiﬁnite trace , and that the symbol M : L1.0, 1/ ! L1.0, 1/ denotes the logarithmic mean Z 1 t ds .M x/.t/ :D x.s/ , x 2 L1.0, 1/, t>0. log.t/ 1 s

In this section, we classify positive Dixmier measurable operators for the Lorentz operator ideal M .M, / associated to a Lorentz function space M (see Exam- ple 2.6.10). The main result of this section is the next theorem.

Theorem 9.2.1. Let : RC ! RC be a concave and increasing function that satis- ﬁes the condition .2t/ lim D 1. (9.2) t!1 .t/

For a positive operator A 2 M .M, / the following conditions are equivalent. (a) A is Dixmier measurable. 274 Chapter 9 Measurability in Lorentz Ideals

(b) There exists the limit Z 1 t lim .s, A/ds. t!1 .t/ 0

It is clear from the theorem that, if 0 A 2 M .M, / is Dixmier measurable, then Z 1 t !.A/ D lim .s, A/ds t!1 .t/ 0 for every Dixmier trace ! on M .M, / (that Dixmier traces exist when satisﬁes the condition (9.2) is given by the results in Chapter 6). To prove the theorem we use the following lemmas.

Lemma 9.2.2. If satisﬁes the condition (9.2), then, for every T>0, we have

.T t/ lim D 1. t!1 .t/

Proof. Without loss of generality, T>1. Select n 2 N such that T 2n.Wehave

.T t/ .2nt/ nY1 .2kC1t/ 1 D . .t/ .t/ .2kt/ kD0

Evidently, .2kC1t/ lim D 1, k 0. t!1 .2kt/

Therefore, .T t/ .T t/ 1 lim inf lim sup 1. t!1 .t/ t!1 .t/

The assertion follows immediately.

Lemma 9.2.3. If satisﬁes the condition (9.2), then, for every increasing function , we have .t/ .t/ D lim sup , D lim inf . t!1 .t/ t!1 .t/

Here, : L1.0, 1/ ! R is given by the formula (6.3). Proof. We prove the ﬁrst assertion only. The proof of the second one is identical. Denote the right-hand side of the ﬁrst formula by C . Section 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals 275

Due to Lemma 9.2.2, for every T>1, we can select N0.T / sufﬁciently large such that 1 1 .N T / 1 C .N /, N>N.T /. T 0

Select N>N0.T / such that .N/ 1 C 1 . .N / T

For every s 2 ŒN , NT,wehave .s/ .N/ .N / .N/ 1 1 1 D C 1 1 C D C 1 . .s/ .N T / .N T / .N / T T T 2

In particular, for every T>1 (and for selected N ), we have Z NT .s/ ds 1 C 1 2 log.T /. N .s/ s T The assertion now follows from the deﬁnition of .

With these results we can prove Theorem 9.2.1. Proof of Theorem 9.2.1. Set Z t .t/ :D .s, A/ds, t>0. 0

By the Hahn–Banach theorem, there exist functionals !1, !2 2 L1.0, 1/ such that ! , ! , ! D , ! D . 1 2 1 2

By Lemma 6.2.7, the functionals !1 and !2 are dilation invariant extended limits. If A is Dixmier measurable, it follows from Lemma 9.2.3 that .t/ lim inf D D !2 D ! .A/ D ! .A/ D !1 t!1 .t/ 2 1 .t/ D D lim sup . t!1 .t/ 276 Chapter 9 Measurability in Lorentz Ideals

9.3 Positive Dixmier Measurable Operators in M1,1

In this section we characterize Dixmier measurable operators in the Dixmier–Macaev operator ideal M1,1.M, /. The function log.1 C t/, t 0, satisﬁes condition (9.2), and therefore the positive Dixmier measurable operators in the Dixmier–Macaev operator ideal satisfy Theorem 9.2.1. As seen in Chapter 8, Dixmier traces on the Dixmier– Macaev operator ideal can be represented by heat kernel functionals and -function residues. Equivalent characterizations of Dixmier measurability, in terms of heat kernel functionals and -function residues are given below. The following theorem is the main result of this section.

Theorem 9.3.1. Let A 2 M1,1.M, / be a positive operator. The following conditions are equivalent. (a) The operator A is Dixmier measurable. (b) There exists the limit Z 1 t lim .s, A/ds. t!1 log.1 C t/ 0

There exists the limit (c) 1 1 lim M .e.A/ / . !1

(d) There exists the limit lim s.A1Cs /. s!0

Furthermore, if any of the conditions (a)–(d) above holds, then we have the coincidence of the three limits Z t 1 1 1 lim .s, A/ds D lim M .e.A/ / D lim s.A1Cs/ t!1 log.1 C t/ 0 !1 s!0 with the value of a Dixmier trace of A. The theorem is proved further below. We ﬁrst discuss the presence of the logarithmic mean M in Theorem 9.3.1. The following corollary follows immediately from Theorem 9.3.1.

Corollary 9.3.2. Let A 2 L1,1.M, / be a positive operator. If there exists the limit

1 1 lim .e.A/ /, !1 Section 9.3 Positive Dixmier Measurable Operators in M1,1 277 then there also exists the limit Z 1 t lim .s, A/ds. t!1 log.1 C t/ 0

The following example shows that we cannot omit the logarithmic mean M from the statement of Theorem 9.3.1 (c).

Example 9.3.3. There exists a positive operator A 2 L1,1.M, / such that Z 1 t lim .s, A/ds D 0 t!1 log.1 C t/ 0 and

1 1 lim sup .e.A/ />0. !1

Proof. Deﬁne a positive operator A by setting 8 n n <ˆs1, s 2 .ee , nee /, n 0 nC1 n nC1 .s, A/ :D ee , s 2 .nee , ee /, n 0 :ˆ e1, s 2 .0, e/.

For every n 0, we have

Z nC1 Z k Z kC1 ee Xn kee ee .s, A/ds D 1 C .s, A/ds C .s, A/ds k k 0 ee kee kD0 Xn k D 1 C log.k/ C 1 .k C 1/e.e1/e kD0 D n C log.nŠ/ C O.1/ D O.nlog.n//.

For every t>e,letn D Œlog.log.t//. It follows that

Z Z nC1 t ee .s, A/ds .s, A/ds D O.nlog.n// D o.log.t//. 0 0 On the other hand, we have

1 Z en 1 X ne X n n 1 1 1 1 1 e 1 e .e.A/ / e sds D e e en e . en nD0 e nD0 278 Chapter 9 Measurability in Lorentz Ideals

n For a given n 2 N,set D ee . It follows that

n n 1 1 1 e 1 e .e.A/ / e e en e D e1 en. Therefore,

1 1 lim sup .e.A/ / e1. !1

To prove Theorem 9.3.1 we start with a well-known Tauberian result. The proof can be found in Hardy [108, Theorem 64]. For this section, the symbol C denotes the Cesàro operator C : L1.0, 1/ ! L1.0, 1/ given by Z 1 t .Cx/.t/ :D x.s/ds, x 2 L1.0, 1/, t>0. t 0

Lemma 9.3.4. Let z 2 L1.0, 1/ be a positive differentiable function and let a 2 R. If tz0.t/ is bounded from below and if .C z/.t/ ! a as t !1, then z.t/ ! a as t !1.

Our next lemma plays an important role in the proof of Theorem 9.3.1.

Lemma 9.3.5. Let z be a positive locally integrable function on .0, 1/ and let a 2 R. 2 If Mz 2 L1.0, 1/ and if .M z/.t/ ! a as t !1, then .M z/.t/ ! a as t !1.

Proof. Set x :D .M z/ ı exp 2 L1.0, 1/. We claim that x satisﬁes the conditions of Lemma 9.3.4. We have Z Z 1 t du 1 log.t/ .M 2z/.t/ D .M z/.u/ D x.s/ds, log.t/ 1 u log.t/ 0 where we have used the substitution u D es in the second equality. By the conditions on z,wehave.Cx/.log.t// ! a as t !1. Equivalently, .Cx/.t/ ! a as t !1. The condition remaining to be shown is that tx0.t/ is bounded from below. We have

Z t Z t 1 e ds 0 1 e ds tx0.t/ D t z.s/ D z.s/ C z.et /. t 1 s t 1 s

0 t Since z is positive, we have tx .t/ .M z/.e / and since Mz 2 L1.0, 1/,we conclude that tx0.t/ const. By Lemma 9.3.4, we have x.t/ ! a as t !1and, therefore, .M z/.t/ ! a as t !1. Section 9.3 Positive Dixmier Measurable Operators in M1,1 279

Lemma 9.3.6. Let x 2 L1.0, 1/ and let a 2 R. The following conditions are equivalent. (a) lim inf x.t/ a lim sup x.t/. t!1 t!1

(b) There exists an extended limit on L1.0, 1/ such that .x/ D a. Proof. The implication (b) ) (a) follows immediately from the deﬁnition of an extended limit. In order to prove the implication (a) ) (b), deﬁne a functional on R C xR by setting .˛ C ˇx/ :D ˛ C ˇa. Clearly,

.z/ lim sup z.t/, z 2 R C xR. t!1

The assertion now follows from the Hahn–Banach theorem.

The following is the classical Karamata theorem. The proof is identical to that of Theorem 8.6.7 and hence it is omitted.

Theorem 9.3.7. Let ˇ be a continuous increasing function. Set Z 1 q h.t/ D e.u=t/ dˇ.u/. 0

We have h.t/ 1 ˇ.t/ lim D 1 C lim t!1 t q t!1 t provided that the left-hand side limit exists. Here is the Gamma function and q>0. We can now prove Theorem 9.3.1.

Proof of Theorem 9.3.1. The implication (b) ) (a) follows from the deﬁnition of !. (a) ) (b). Suppose that !.A/ D a for every dilation invariant extended limit !. It follows from Lemma 7.4.1 that ıM .A/ D a for every extended limit .Thatis, we have the equality Z 1 t . ı M/ .s, A/ds D a. log.1 C t/ 0 It follows from Lemma 9.3.6 that Z 1 t lim M .s, A/ds D a. (9.3) t!1 log.1 C t/ 0 280 Chapter 9 Measurability in Lorentz Ideals

Set z.t/ :D t.t, A/. Observe that z is a positive measurable, but not necessarily bounded, function. Note, however, that Z 1 t .M z/.t/ D .s, A/ds. log.t/ 1

Since A 2 M1,1.M, / it follows that Mz 2 L1.0, 1/. Observe that Z 1 t lim .M z/.t/ .s, A/ds D 0. (9.4) t!1 log.1 C t/ 0

Recall that .My/.t/ ! 0ast !1whenever y 2 L1.0, 1/ is such that y.t/ ! 0 as t !1. Combining (9.3) and (9.4), we infer that .M 2z/.t/ ! a as t !1.By Lemma 9.3.5, we infer that .M z/.t/ ! a as t !1. The proof of the implication is completed by referring to (9.4). (c) ) (a). Let a be the limit in (c). By deﬁnition of the functional !,wehavethat !.A/ D a for every dilation invariant extended limit !. By Theorems 6.4.1 and 8.3.6, the set of all Dixmier traces coincides with that of all heat kernel functionals. Hence, !.A/ D a for every dilation invariant extended limit !. (a) ) (c). Suppose that !.A/ D a for every dilation invariant extended limit !. The same argument as above shows that !.A/ D a for every dilation invariant extended limit !. It follows from Lemma 7.4.1 that ıM .A/ D a for every extended limit .Thatis, 1 1 . ı M 2/ .e.A/ / D a. It follows from Lemma 9.3.6 that 1 1 lim M 2 .e.A/ / D a. t!1 We know that the mapping 1 1 ! M .e.A/ / is bounded. The proof of the implication is completed by using Lemma 9.3.5. (a) ) (d). Suppose that !.A/ D a for every dilation invariant extended limit !. It follows from Theorems 8.6.4 and 6.4.1 that the set of all -function residues is a subset of the set of Dixmier traces. Hence, for every extended limit ,wehave 1 .A1C1=t/ D a. t

Applying Lemma 9.3.6 completes the proof of the implication. Section 9.4 C -invariant Extended Limits 281

(d) ) (a). Without loss of generality, kAk1 1. By assumption, we have 1 lim .A1C1=r/ D a. r!1 r Deﬁne a continuous increasing function ˇ on .0, 1/ by setting Z u v v ˇ.u/ :D e dnA.e /. 0 It is clear that Z Z Z 1 1 1 u=r u.1C1=r/ u 1C1=r h.r/ :D e dˇ.u/ D e dnA.e / D dnA./ 0Z 0 0 1 1C1=r 1C1=r D dnA./ D .A /. 0 It follows from Theorem 9.3.7 that h.r/ ˇ.u/ a D lim D lim . r!1 r u!1 u Therefore, Z Z ˇ.u/ 1 1 1 1 a D lim D lim dnA./ D lim dnA./. !1 !1 u u u u u e t!1 log.1 C t/ 1=t

Hence, Z 1 1 lim dnA./ D a. t!1 log.1 C t/ log.t/=t

The assertion now follows from Theorem 7.2.1.

9.4 C -invariant Extended Limits

Deﬁne the Cesàro operator C : L1.R/ ! L1.R/ by Z 1 t .Cx/.t/ :D x.s/ds, x 2 L1.R/, t 2 R. t 0 This is a slight extension of the operator C from the last section (and Section 4.5) in that tR can takeR negative values (and the order of integration is reversed in that case so t 0 that 0 :D t , t<0). The main objective of this section is to characterize the range !.x/ where x 2 L1.R/ is real-valued and ﬁxed and ! varies over all C -invariant extended limits on 282 Chapter 9 Measurability in Lorentz Ideals

L1.R/. By an extended limit on L1.R/ we mean a state on the algebra L1.R/ that vanishes on functions that have support in an interval .1, t,forsomet 2 R.The main result is given in Theorem 9.4.3 and a simpler criterion is given in Theorem 9.4.9. The latter result is used in Theorem 9.5.1 in the next section to present a description of the set of all positive M -measurable operators.

Lemma 9.4.1. For every real-valued x 2 L1.R/ we have

lim sup.Cx/.t/ lim sup x.t/. t!C1 t!C1

Proof. It is sufﬁcient to prove the assertion for x Ckxk1 instead of x. Therefore, we may assume that x 0. For every ﬁxed t0 > 0wehave

lim sup.Cx/.t/ D lim sup.Cx.t0,1//.t/ sup.Cx/.t/ sup x.t/. t!C1 t!C1 tt0 tt0

If t0 !C1, the right-hand side converges to lim supt!C1 x.t/.

Lemma 9.4.2. For every real-valued x 2 L1.R/, the following limit exists

1 1 Xn p.x/ :D lim lim sup C kx .t/. (9.5) n!1 n t!C1 kD0 So deﬁned, the functional p is convex and homogeneous.

Proof. Set Xn1 k ˛n :D lim sup C x .t/, n 2 N. t!C1 kD0

We have

Xn1 mX1 k n k ˛nCm D lim sup C x .t/ C C C x .t/ t!C1 kD0 kD0 Xn1 mX1 lim sup C kx .t/ C lim sup C n C kx .t/. t!C1 t!C1 kD0 kD0

It follows from Lemma 9.4.1 that

˛nCm ˛n C ˛m.

The existence of the limit follows from Lemma 3.4.6. Section 9.4 C -invariant Extended Limits 283

The next theorem is the main technical result of this section.

Theorem 9.4.3. Let x 2 L1.R/ be real-valued and let a 2 R. The following conditions are equivalent. 1 Xn 1 1 Xn 1 (a) lim lim inf C kx .t/ a lim lim sup C kx .t/. n!1 n t!C1 n!1 n t!C1 kD0 kD0

(b) There exists an extended limit ! on L1.R/ such that ! D ! ı C and !.x/ D a.

Proof. (a) ) (b). Let p be the functional deﬁned in Lemma 9.4.2. We have p.x/ a p.x/. Deﬁne the functional ! on R C xR by the formula

!.1 C 2x/ D 1 C 2a, 1, 2 2 R.

We have that ! p on R C xR.Indeed,for1, 2 2 R,

!.1 C 2x/ D 1 C 2a 1 Cj2jp.sgn.2/x/ D p.1 C 2x/.

By the Hahn–Banach theorem, the functional ! can be extended to L1.R/ preserving the inequality ! p. If z 0then.p.z// !.z/ p.z/. It follows that ! is a positive functional since !.z/ p.z/ 0. It also follows that !.1/ D 1since1 Dp.1/ !.1/ p.1/ D 1. Hence, ! is a state. Suppose that z.t/ ! 0ast !1. It follows that, for every k 2 N, .C kz/.t/ ! 0 as t !1. Hence, for every n 2 N,wehave

Xn1 lim C kz .t/ D 0 t!1 kD0 and, therefore, p.z/ D p.z/ D 0. Thus, !.z/ D 0. It follows that ! is an extended limit. Note that Xn1 C k.1 C/D 1 C n. kD0

Therefore, 1 p.z Cz/D lim lim sup.z C nz/ D 0. n!1 n t!1

(b) ) (a). Let x 2 L1.R/ be real-valued and let ! be a C -invariant extended limit. We have 1 1 1 Xn 1 Xn !.x/ D ! C kx lim sup C kx .t/. n n t!1 kD0 kD0

Taking the limit n !1, we conclude that !.x/ p.x/. It follows that p.x/ !.x/ p.x/.

Theorem 9.4.3 can be used to describe the positive M -measurable operators. How- ever, the following lemmas lead to a much simpler description.

Lemma 9.4.4. For every real-valued x 2 L1.R/, the following limit exists

q.x/ D lim lim sup.C nx/.t/. n!1 t!1

Proof. The assertion follows from Lemma 9.4.1.

Lemma 9.4.5. For every real-valued x 2 L1.R/, we have p.x/ q.x/.

Proof.Forn 0set n ˛n :D lim sup.C x/.t/. t!1

Evidently, Xn1 Xn1 k lim sup C x .t/ ˛k, n 0. t!1 kD0 kD0

Therefore, 1 1 Xn q.x/ D lim ˛n D lim ˛k p.x/. n!1 n!1 n kD0

The proof of the converse inequality is more complicated.

Lemma 9.4.6. Let 0 x 2 L1.R/. For every >0, there exists n 2 N and (an n arbitrary large) t0 > 0 such that .C x/.t/ .1 /q.x/ for every t 2 Œt0, t0=4.

Proof. Without loss of generality, q.x/ > 0. Fix 0 <<1 and choose n 2 N such that 1 lim sup.C n1x/.t/ q.x/ 1 C 2 . t!1 2 Section 9.4 C -invariant Extended Limits 285

n1 2 Hence, there exists t1 > 0 such that .C x/.t/ q.x/.1 C / for every t t1.If 1 lim inf.C nx/.t/ 1 q.x/, t!1 2 then the assertion is evident. Suppose then that 1 lim inf.C nx/.t/ < 1 q.x/, t!1 2 and note that lim sup.C nx/.t/ q.x/. t!1

n 2 Select (an arbitrarily large) t2 such that .C x/.t2/ D .1 /q.x/.Let 1 t D sup t 2 Œt , t : .C nx/.t/ D 1 q.x/ . 0 1 2 2

It follows that Z Z Z t2 t0 t2 2 n1 n1 n1 .1 /t2q.x/ D .C x/.s/ds D .C x/.s/ds C .C x/.s/ds 0 0 t0 t .C nx/.t / C .t t /.1 C 2/q.x/ 0 0 2 0 D t 1 q.x/ C .t t /.1 C 2/q.x/. 0 2 2 0

Hence, 1 .1 2/t t 1 C .t t /.1 C 2/. 2 0 2 2 0

Simple arithmetic gives us that t0 4t2. The assertion now follows from the deﬁni- tion of t0.

Lemma 9.4.7. Let 0 x 2 L1.R/. For every natural number r 2, there exist m 2 N and (an arbitrary large) t 2 R such that 2 .C mCsx/.t/ 1 q.x/,0 s

Proof. Due to Lemma 9.4.6, there exists m 2 N and (an arbitrary large) n 2 N such that 1 .C mx/.t/ q.x/ 1 t 2 Œn, r 2r n. (9.7) r 2 286 Chapter 9 Measurability in Lorentz Ideals

We claim that, for every natural number 0 s

The inequality (9.8) holds for s D 0 because of (9.7). We use an argument by induction. Suppose that (9.8) holds for some 0 s

Hence (9.8) is shown for s C 1, and the claim is shown. It follows from (9.7) that, for t D r 2r n and 0 s

Lemma 9.4.8. For every real-valued x 2 L1.R/, we have p.x/ q.x/.

Proof. It is sufﬁcient to prove the assertion for x Ckxk1 instead of x. Therefore, without loss of generality, we assume that x 0. Fix r 2 N. By Lemma 9.4.7, there exists m 2 N and an arbitrary large t such that Xr1 C mCkx .t/ .r 2/q.x/. kD0

Therefore, Xr1 lim sup C mCkx .t/ .r 2/q.x/. (9.9) t!1 kD0

It follows from (9.9) and Lemma 9.4.1 that

1 Xr 1 1 Xr 1 2 lim sup C kx .t/ lim sup .C mCkx/.t/ 1 q.x/. r t!1 r t!1 r kD0 kD0

Taking the limit r !1, the assertion follows.

Theorem 9.4.9. Let x 2 L1.R/ be real-valued and let a 2 R. The following conditions are equivalent. Section 9.5 Positive M -measurable Operators 287

(a) lim lim inf.C nx/.t/ a lim lim sup.C nx/.t/. n!1 t!1 n!1 t!1

(b) There exists an extended limit ! on L1.R/ such that ! D ! ı C and !.x/ D a.

Proof. The assertion follows from Theorem 9.4.3, Lemma 9.4.5 and Lemma 9.4.8.

0 Let L1.0, 1/ L1.0, 1/ denote the subspace of (equivalence classes of) bounded functions that vanish at inﬁnity.

Corollary 9.4.10. Let x 2 L1.0, 1/ be real-valued and let a 2 R such that !.x/ D n a for every extended limit ! on L1.R/ such that ! D ! ıC . It follows that C x ! a 0 in the Calkin algebra L1.0, 1/=L1.0, 1/.

Proof. It follows from Theorem 9.4.9 that

lim lim sup.C nx/.t/ D a. n!1 t!1

n 0 Hence, .C x a/C ! 0 in the Calkin algebra L1.0, 1/=L1.0, 1/. Similarly, n .C x a/ ! 0 in the Calkin algebra. Therefore,

n n n C x a D .C x a/C .C x a/ ! 0 in the Calkin algebra.

9.5 Positive M-measurable Operators

We are now able to characterize the positive M -measurable operators in the Dixmier– Macaev operator ideal M1,1.M, /. The characterization given in Theorem 9.5.1 below, and the characterization of Dixmier measurable operators given in Theorem 9.3.1, allows us to show that there is a positive M -measurable operator which is not Dixmier measurable (Theorem 9.5.4). Our technique is to transfer the results about C -invariant extended limits on L1.R/ from the last section to M -invariant extended limits on L1.0, 1/. We do this using the exponential map exp : R ! .0, 1/ or, equivalently, exp : L1.0, 1/ ! L1.R/ where exp.x/ :D x ı exp, x 2 L1.0, 1/. For every t>0, and x 2 L1.0, 1/,we have

Z t Z 1 e ds 1 t ..M x/ ı exp/.t/ D x.s/ D x.eu/du D C.x ı exp/.t/. t 1 s t 0 R R b a When t<0 the same identity equality holds, understanding that a :D b if b

an extended limit ! on L1.R/ is C -invariant if and only if the extended limit ! ı exp on L1.0, 1/ is M -invariant, since

.! ı exp/.M x/ D .! ı C/.exp.x//, x 2 L1.0, 1/.

0 In this section, L1.0, 1/ L1.0, 1/ is the subspace of (equivalence classes of) bounded functions that vanish at inﬁnity.

Theorem 9.5.1. A positive operator A 2 M1,1.M, / is M -measurable if and only if Z 1 t M n .s, A/ds ! const log.1 C t/ 0

0 in the Calkin algebra L1.0, 1/=L1.0, 1/.

Proof. An extended limit ! on L1.R/ is C -invariant if and only if the extended limit ! ı exp on L1.0, 1/ is M -invariant. Hence, A is M -measurable if and only if there exists a 0 such that

Z t 1 e ! t .s, A/ds D a, ! D ! ı C . log.1 C e / 0 It follows from Corollary 9.4.10 that Z et n 1 C t .s, A/ds ! a log.1 C e / 0

0 in the Calkin algebra L1.0, 1/=L1.0, 1/. The assertion follows immediately.

We use the characterization in Theorem 9.5.1 to show that there is an M -measurable positive operator in M1,1.M, /which is not Dixmier measurable. Consider the following operator K : L2.1, e/ ! L2.1, e/ Z Z 1 t 1 e .Kx/.t/ :D x.s/ds C x.s/ds, x 2 L2.1, e/. t 1 t.e 1/ 1

In the rest of this section we use the notation xn x to denote uniform convergence in CŒ1, e.

Lemma 9.5.2. For every x 2 CŒ1, e, we have Z e ds Knx x.s/ . 1 s Section 9.5 Positive M -measurable Operators 289

Proof. The operator K : L2.1, e/ ! L2.1, e/ is Hilbert–Schmidt and is, therefore, compact. Hence, the (non-zero part of the) spectrum .K/ of K consists of discrete eigenvalues with ﬁnite algebraic mutliplicity. We claim that .K/ Df0g[f1g and 1 is a simple eigenvalue. Indeed, let ¤ 0be an eigenvalue and let x 2 L2.1, e/ be the corresponding eigenfunction. We have Z Z 1 t 1 e x.s/ds C x.s/ds D x.t/. (9.10) t 1 t.e 1/ 1

Since x 2 L1.1, e/, the left-hand side depends continuously on t. Thus, x 2 CŒ1, e. Also the left-hand side is a continuously differentiable function of t and, therefore, so is x. Multiplying both sides of (9.10) by t and taking derivatives, we obtain that

x.t/ D .tx.t//0.

Therefore, x.t/ D Ct.1/=. Substituting into (9.10), we obtain D 1 and, therefore, x D 1. This proves the claim. 1 Let z.t/ D t for t 2 Œ1, e and let L be the hyperplane in L2.1, e/ orthogonal to z. Note that Kz D z. Therefore,

.Kx, z/ D .x, Kz/ D .x, z/ D 0, x 2 L.

Hence, K : L ! L.Since1… L, it follows that .K : L ! L/ Df0g and, therefore, n K : L ! L is a quasi-nilpotent operator. Hence, K x ! 0inL2.1, e/ for every x 2 L. Evidently,

kxk1 D K.kxk1/ Kx K.kxk1/ Dkxk1.

0 Therefore, K : CŒ1, e ! CŒ1, e is a contraction. Observe that k.Kx/ k1 .e C 1/kxk1 for every x 2 CŒ1, e. Hence, the image of the unit ball under K is an equicon- tinuous set. It follows from the Arzela–Ascoli theorem that K : CŒ1, e ! CŒ1, e is a compact operator. Let x 2 CŒ1, e \ L and let z be a limit point of the sequence Knx, n 0. Since n n K x ! 0inL2.1, e/, it follows that z D 0. Thus, K x ! 0inCŒ1, e. For every x 2 CŒ1, e,wehavex .x, z/ 2 CŒ1, e \ L. Therefore, Kn.x .x, z// 0. Since K1 D 1, the assertion follows.

Deﬁne the mapping : .0, 1/ ! Œ1, e by setting

.t/ :D teŒlog.t/, t>0. (9.11) 290 Chapter 9 Measurability in Lorentz Ideals

Lemma 9.5.3. For every x 2 CŒ1, e, we have Z e ds C n.x ı / x.s/ . 1 s Proof. Fix t>0andsetk :D Œlog.t/. It is clear that

Z iC1 Z 1 X e t .C.x ı //.t/ D x.ei s/ds C x.eks/ds t ei ek i

Therefore, C.xı / D .Kx/ ı . By induction, we have C n.x ı / D .Knx/ı .The assertion now follows from Lemma 9.5.2.

Theorem 9.5.4. There exists a positive operator A 2 M1,1.M, / which is M -measurable, but not Dixmier measurable.

Proof. Choose a positive operator A such that

kek .A/ D sup e ek . k0 Œ0,e

k kC1 For every t 2 Œee , ee ,wehave

Z Z k t ee k e .s, A/ds D .s, A/ds C.t ee /.t, A/ D ek Ct.t, A/CO.1/. 0 0 e 1

In particular, A 2 M1,1.M, /. We have Z t 1 e k .s, A/ds D C o.1/, t D ee , log.1 C t/ e 1 Z0 t 1=2 1 e kC1=2 .s, A/ds D C o.1/, t D ee . log.1 C t/ 0 e 1

Clearly, A fails the condition in Theorem 9.3.1 (b) and, therefore, it is not Dixmier measurable. For every dilation invariant extended limit ! on L1.0, 1/,wehave

1 t.t, A/ e X ek .A/ D ! C ! ek ekC1 .t/ . ! log.1 C t/ e 1 log.t/ Œe ,e kD0 Section 9.6 Additional Invariance of Dixmier Traces 291

It follows from Lemma 7.2.4 (a) that

1 e X ek .A/ D ! ek ekC1 .t/ . ! e 1 log.t/ Œe ,e kD0

Set z.t/ D t 1 for every t 2 Œ1, e and let : .0, 1/ ! .0, 1/ be as in (9.11). It is clear that 1 X ek ek ekC1 .t/ D .z ı /.log.t//. log.t/ Œe ,e kD1 Therefore, e .A/ D .! ı log/.z ı /. ! e 1 Suppose now that ! D ! ı M . From the argument at the beginning of the section, we obtain that ! ı log D .! ı log/ ı C is a C -invariant extended limit on L1.R/.It follows from Lemma 9.5.3 that Z e ds C n.z ı / z.s/ D 1. 1 s Therefore, for every M -invariant extended limit !, e e e .A/ D .! ı log/.z ı / D .! ı log/.C n.z ı // ! , ! e 1 e 1 e 1 and the positive operator A is M -measurable.

9.6 Additional Invariance of Dixmier Traces

0 In this section, L1.0, 1/ L1.0, 1/ is the subspace of (equivalence classes of) bounded functions vanishing at inﬁnity, and C0.0, 1/ is the subspace of continuous functions vanishing at inﬁnity.

Lemma 9.6.1. The operator MC M maps L1.0, 1/ into C0.0, 1/.

Proof. For every t>1, we have Z Z 1 t 1 s ds .MCx/.t/ D x.u/du log.t/ 1 s 0 s Z Z Z Z 1 1 t ds 1 t t ds D x.u/ 2 du C x.u/ 2 du log.t/ 0 1 s log.t/ 1 u s Z Z 1 1 1 1 t x.u/ D 1 x.u/du C du log.t/ t 0 log.t/ 1 u Z 1 1 t x.u/du log.t/ t 1 Z 1 1 1 D x.u/du C .M x/.t/ .Cx/.t/. log.t/ 0 log.t/ Hence, the function Z 1 1 1 .MCx M x/.t/ D x.u/du .Cx/.t/ (9.12) log.t/ 0 log.t/ belongs to C0.0, 1/. Although we do not need the result of the following lemma, we present it here for the sake of completeness.

0 Lemma 9.6.2. The operator CM M maps L1.0, 1/ into L1.0, 1/. Proof. For every t>1, we have Z Z Z 1 1 1 t 1 s du .CM x/.t/ D .M x/.s/ds C x.u/ ds t 0 t 1 log.s/ 1 u Z Z Z 1 1 1 t x.u/ t ds D .M x/.s/ds C du. t 0 t 1 u u log.s/ However, Z t ds D li.t/ li.u/, u log s Section 9.6 Additional Invariance of Dixmier Traces 293 where li is the logarithmic integral function deﬁned by the formula Z t ds li.t/ :D . e log.s/ Therefore, Z Z Z 1 1 li.t/ t du 1 t li.u/ .CM x/.t/ D .M x/.s/ds C x.u/ x.u/ du t 0 t 1 u t 1 u Z Z 1 1 li.t/ log.t/ 1 t li.u/ D .M x/.s/ds C .M x/.t/ x.u/ du. t 0 t t 1 u For the function li, we have the following asymptotic formula t li.t/ D .1 C o.1//. (9.13) log.t/

Therefore, for every t>e,wehave ˇ Z ˇ Z ˇ t ˇ t ˇ li.u/du ˇ jli.u/jdu ˇ x.u/ ˇ kxk1 Dkxk1.log.t/ li.t/ t C 2e 1/. 1 u 1 u It now follows from (9.13) that Z 1 t li.u/ lim x.u/ du D 0. t!1 t 1 u Therefore, 0 .CM x/ 2 .1 C o.1// Mx C L1.0, 1/

0 for every x 2 L1.0, 1/. It follows that CM M maps L1.0, 1/ into L1.0, 1/.

An important corollary of Lemma 9.6.1 is:

Corollary 9.6.3. Every M -invariant extended limit on L1.0, 1/ is also C -invariant.

Proof.Let! be an M -invariant extended limit and let x 2 L1.0, 1/. It follows from Lemma 9.6.1 that

!.Cx/ D !.MCx/ D !.Mx/ D !.x/.

Lemma 9.6.4. For every x 2 L1.0, 1/, the function .Cx/ ı exp is uniformly continuous. 294 Chapter 9 Measurability in Lorentz Ideals

Proof. It is clear that Z t Z t e 0 e ..Cx/.et //0 D et x.s/ds Det x.s/ds C x.et /. 0 0 Therefore, 0 ..Cx/ ı exp/ D.Cx/ ı exp Cx ı exp 2 L1.0, 1/. Hence, .Cx/ ı exp is a Lipschitz function and, therefore, is uniformly continuous.

Lemma 9.6.5. Let s ! hs be a norm-continuous map from the interval Œa, b into L1.0, 1/. If ! 2 L1.0, 1/ , then the function s ! !.hs/ is Riemann-integrable and Z Z b b ! hsds D !.hs /ds. a a Proof. Fix >0 and select n 2 N such that b a kh h k1 , 8s , s > 0: js s j . s1 s2 1 2 1 2 n It follows that Z b Xn1 b a hsds h C .b a/. n a i.b a/=n a iD0 1 Thus, n1 Z b a X b h C h ds n a i.b a/=n s iD0 a as n !1. It follows that Z b a Xn 1 b !.h C / ! ! h ds . n a i.b a/=n s iD0 a Evidently,

j!.hs1 / !.hs2 /jk!kL1khs1 hs2 k1.

Hence, the mapping s ! !.hs/ is continuous and, therefore, Riemann-integrable. Hence, Z b a Xn 1 b !.h C / ! !.h /ds. n a i.b a/=n s iD0 a The assertion follows immediately. Section 9.6 Additional Invariance of Dixmier Traces 295

Lemma 9.6.6. Let ! be a dilation invariant extended limit on L1.0, 1/. For every x 2 L1.0, 1/ such that x ı exp is uniformly continuous, we have

!.x/ lim sup sup.M1=t x/.s/. s!1 t>0

Proof. Fix s>0. Since x ı exp is uniformly continuous, it follows that the mapping u ! 1=ux is norm-continuous on the interval Œ1, s. Therefore, Z Z s du s du ! 1=ux D !.1=ux/ D !.x/ log.s/. 1 u 1 u

Note that .1=ux/.t/ D .1=tx/.u/. Therefore, Z Z 1 s du 1 s du !.x/ D ! 1=ux sup .1=tx/.u/ log.s/ 1 u t>0 log.s/ 1 u

D sup.M1=t x/.s/. t>0 Taking the limit s !1, we obtain the assertion.

Lemma 9.6.7. Let x 2 L1.0, 1/ be such that x ı exp is uniformly continuous. For every dilation invariant extended limit ! on L1.0, 1/ and for every n 2 N, we have !.x/ D !.C nx/.

Proof. By Lemma 9.6.6, we have

!.x Cx/ lim sup sup..M1=t /.x Cx//.s/. s!1 t>0

However, 1=t commutes with C . Therefore,

!.x Cx/ lim sup sup..M.1 C //.1=tx//.s/. s!1 t>0 It follows from (9.12) that .C z/.1/ .C z/.s/ ..MC M /z/.s/ D . log.s/ Hence,

2kzk1 j..MC M /z/.s/j . j log.s/j

Substituting z D 1=tx, we obtain

2kxk1 !.x Cx/ lim sup D 0. s!1 j log.s/j Substituting x !x, we obtain !.Cx x/ 0. Therefore, !.x/ D !.Cx/. 296 Chapter 9 Measurability in Lorentz Ideals

By Lemma 9.6.4, .C kx/ ı exp is uniformly continuous for every k 2 N. Note that

Xn1 x C nx D .1 C/z, z D C kx. kD0

Applying the result of the previous paragraph to z, we obtain !.x C nx/ D !.z Cz/D 0. The assertion follows immediately.

Lemma 9.6.8. For every dilation invariant extended limit ! on L1.0, 1/, there exists a dilation and C -invariant extended limit !0 on L1.0, 1/ such that !.x/ D !0.x/ whenever x 2 L1.0, 1/ is such that x ı exp is uniformly continuous. Proof. Let ! be a dilation invariant extended limit and let E be the set of all x 2 L1.0, 1/ such that x ı exp is uniformly continuous. We apply Theorem 6.2.5 to the commutative semigroup

n G DfC 1=t : n 0, t>0g, the linear subset E L1.0, 1/, the linear functional !jE and the convex functional lim sup . The restriction of ! on E can be extended to the dilation and C -invariant functional !0 2 L1.0, 1/ .Since

!0.z/ lim sup z.t/, z 2 L1.0, 1/, t!1 it follows that !0 is an extended limit. Recall that ! D !0 on E by the deﬁnition of !0. The assertion is proved.

The following theorem proves that, in the construction of a Dixmier trace ! on M1,1.M, / given in Section 6.3 (see Theorem 6.3.6), the extended limit ! on L1.0, 1/ can be assumed to be both dilation and C -invariant.

Theorem 9.6.9. Let ! be a dilation invariant extended limit on L1.0, 1/ and let ! denote the Dixmier trace on M1,1.M, /associated to !. For any dilation invariant extended limit ! on L1.0, 1/ there exists a dilation and C -invariant extended limit

!0 on L1.0, 1/ such that ! D !0 . Proof. We claim that the mapping Z t 1 e t ! t .s, A/ds, t>0, log.1 C e / 0 Section 9.7 Measurable Operators in L1,1 297 is a Lipschitz function (and, therefore, is uniformly continuous). We have,

Z t 1 e 0 t .s, A/ds log.1 C e / 0 Z t et .et , A/ et 1 e D t t 2 .s, A/ds. log.1 C e / .1 C e / log .1 C et / 0 Clearly,

Z t et .et , A/ 1 e M 0 t t .s, A/ds kAk 1,1 log.1 C e / log.1 C e / 0 and Z t 1 e 1 M 2 .s, A/ds kAk 1,1 . log .1 C et / 0 log.2/ Hence, ˇ Z ˇ ˇ et 0ˇ ˇ 1 ˇ 1 M ˇ t .s, A/ds ˇ 1 C kAk 1,1 log.1 C e / 0 log.2/ and the claim follows. The assertion now follows from Lemma 9.6.8.

9.7 Measurable Operators in L1,1

The sections of this chapter have characterized the positive Dixmier measurable and positive M -measurable operators in the Dixmier–Macaev operator ideal M1,1.M, /. We show in this section that Dixmier measurability of arbitrary operators in the ideal L1,1.L.H /,Tr/ of compact operators can be characterized by eigenvalues.

Deﬁnition 9.7.1. An arbitrary compact operator A in the Lorentz ideal M1,1.L.H /,Tr/ is called Tauberian if the limit

1 Xn lim .j , A/ n!1 log.2 C n/ j D0 exists, where .A/ is an eigenvalue sequence of A.

It follows from Theorem 9.3.1 that every positive operator A 2 M1,1.L.H /,Tr/ is Dixmier measurable if and only if it is Tauberian, since

1 Xn lim .j , A/ n!1 log.2 C n/ j D0 298 Chapter 9 Measurability in Lorentz Ideals

exists. Theorem 9.7.5 below shows that an arbitrary operator A 2 L1,1.L.H /,Tr/ is Dixmier measurable if and only if it is Tauberian. This result cannot be extended to arbitrary compact operators in the Dixmier–Macaev ideal M1,1.L.H /,Tr/.Ex- ample 9.7.6 provides an example of an arbitrary Dixmier measurable operator A 2 M1,1.L.H /,Tr/ that is not Tauberian.

Let X0 L1.0, 1/ be the linear span of the functions Z 1 t t ! .s, A/ds, A 2 L1,1.L.H /,Tr/. log.1 C t/ 0

Lemma 9.7.2. If ! is an extended limit on L1.0, 1/,then!.ux/ D !.x/ for every x 2 X0 and for every u>0.

Proof. Since u1=u is the identity operator, it is sufﬁcient to prove the assertion for u 2 .0, 1/ and for Z 1 t x.t/ D .s, A/ds, A 2 L1,1.L.H /,Tr/. log.1 C t/ 0

It follows from the deﬁnition of L1,1.L.H /,Tr/ that Z 1 t=u .ux/.t/ D .s, A/ds log.1 C t=u/ 0 Z 1 t t.t, A/ D .s, A/ds C O log.1 C t=u/ 0 log.1 C t/ Z 1 t D .s, A/ds C o.1/. log.1 C t=u/ 0 The assertion now follows from Remark 6.3.5.

0 Let X be the linear span of uX0, u>0, and L1.0, 1/.

Lemma 9.7.3. The space X is dilation invariant, that is uX D X, u>0. For every extended limit ! on L1.0, 1/, we have !.ux/ D !.x/ for every x 2 X and every u>0.

Proof. The ﬁrst assertion is trivial. The second assertion follows from Lemma 9.7.2.

Lemma 9.7.4. For every extended limit ! on L1.0, 1/, the functional ! : L1,1.L.H /,Tr/C ! R (as in Deﬁnition 6.3.2) is additive and extends to a Dixmier trace on M1,1.L.H /,Tr/. Section 9.7 Measurable Operators in L1,1 299

Proof. It follows from Theorem 6.2.5 that there is a dilation invariant extension !0 of the functional !jX satisfying the condition

!.x/ lim sup x.t/, x 2 L1.0, 1/. t!1

That is, !0 is a dilation invariant extended limit on L1.0, 1/. The assertion now follows from Theorem 6.3.3.

The following theorem characterizes Dixmier measurable operators in L1,1.L.H /,Tr/.

Theorem 9.7.5. Let A 2 L1,1.L.H /,Tr/. The following conditions are equivalent. (a) A is Dixmier measurable.

(b) A is Tauberian, that is the limit

1 Xn lim .k, A/ n!1 log.2 C n/ kD0 exists, where .A/ is an eigenvalue sequence of A.

Proof. Suppose ﬁrst that A 2 L1,1.L.H /,Tr/ is self-adjoint. Set Z 1 t a1 D lim inf ..s, AC / .s, A//ds, t!1 log.1 C t/ Z0 1 t a2 D lim sup ..s, AC / .s, A//ds. t!1 log.1 C t/ 0

By Lemma 9.3.6, we can select extended limits !1, !2 such that Z 1 t ai D !i ..s, AC/ .s, A//ds , i D 1, 2. .t/ 0 L L R Thus, !i .x/ D ai . It follows from Lemma 9.7.4 that !i : 1,1. .H /,Tr/C ! extend to Dixmier traces on M1,1.L.H /,Tr/. Thus, if A is Dixmier measurable, then a1 D !1 .A/ D !2 .A/ D a2. Consequently, there exists the limit Z 1 t lim ..s, AC / .s, A//ds t!1 log.1 C t/ 0 or, equivalently, the limit

1 Xn lim ..k, AC/ .k, A// . n!1 log.2 C n/ kD0 300 Chapter 9 Measurability in Lorentz Ideals

By Lemma 5.7.1,

Xn Xn ..k, AC/ .k, A// D .k, A/ C O.1/ kD0 kD0 and the assertion follows. Consider now the general case. An operator A is Dixmier measurable if and only if both

Example 9.7.6. There exists a Dixmier measurable operator A 2 M1,1.L.H /,Tr/ that is not Tauberian.

Proof. Let A D A 2 M1,1.L.H /,Tr/ be such that 2.A/ D 2.AC/.For example,

n n n2 n n12 n .AC/ :D sup 2 Œ0,22 /, .A/ :D sup 2 Œ0,21C2 /. n0 n0 An easy computation shows that A is not Tauberian. Observe that A is Dixmier measurable since every Dixmier trace vanishes on A.

9.8 Notes

There are several natural subsets of Dixmier traces considered in noncommutative geometry. Dixmier’s original article [62] dealt with the set of traces Tr! where ! is a translation and dilation invariant extended limit on l1. Alain Connes in his famous article [45] initially considered precisely this class of limits (termed in [45] the ‘class of all means on the amenable group of upper triangular two by two matrices’). Later, it was observed by Connes [48] that in order to ensure that Tr! is additive on M1,1.L.H /,Tr/C, it is sufﬁcient only to assume that ! is a dilation invariant extended limit on l1. In [153], the set D of all traces generated by such dilation invariant extended limits was termed the set of Dixmier traces and this terminology has been adopted in this book. It follows from Theorem 9.6.9, that the set of traces introduced by Dixmier in [62] actually coincides with the one introduced by Connes (which is the set of all Dixmier traces). Connes suggestedanother way to generate dilation invariantextendedlimits [48, SectionIV, 2ˇ] by observing that for any extended limit on l1 the functional ! :D ı M is dilation invariant. Here, the bounded operator M : l1 ! l1 is given by the formula 1 Xn x .Mx/ :D k , n 1. n log.1 C n/ k kD1 Section 9.8 Notes 301

The set C of all Dixmier traces ! defined this way were termed the set of Connes–Dixmier traces in [153] and we refer the reader to that paper for additional information. Finally, there is the set B of all Dixmier traces defined by M -invariant extendedlimits. The- orem 9.5.4 and results in [153] indicate that the set B is a proper subset of C and D. Various important formulas in noncommutative geometry, like Connes’ formula for the representative of the Hochschild class of the Chern character for .p, 1/-summable spectral triples (see e.g. [29, Theorem 7] and [9, Theorem 6]), as well as those involving heat kernel functionals and generalized -functions (see e.g. [9, 29, 32, 33, 240]), are established for Dixmier traces defined by M -invariant extended limits. Dixmier traces were first considered for general Lorentz ideals in [67,103]. Conditions for the additivity of ! were studied in the context of fully symmetric functionals in [9, 65–67], where various classes of extended limits on l1 with additional invariance properties (and corresponding subsets of Dixmier traces) were introduced. The idea to relate various questions concerning measurability of operators from M1,1.M, /with the classical studies of G. G. Lorentz [155] on almost convergent sequences was presented in [66] (see also [153] where the first attempt to classify measurable elements in Lorentz spaces was made). Our present approach is mainly borrowed from [236] and consists in using the technique invented by L. Sucheston [228], who significantly simplified the original proofs of Lorentz. Theorem 9.3.1 has its roots in [35]. We refer to the papers [235–237] for additional studies of various subsets of Dixmier traces that were not included in this chapter. In particular, [236] contains criteria for the Dixmier measurability of not necessarily positive operators. The materialin Section9.7 has been adoptedfrom the paper [214]. Example 9.7.6 shows that the analog of Theorem 9.7.5 does not hold for M1,1.L.H /,Tr/, that is the set of all Dixmier measurable elements in M1,1.L.H /,Tr/ is strictly larger than the set of all Tauberian elements from that space.For a description of the former set we refer to [236]. The paper [214] also contains a complete characterization of the largest symmetric subspace of M1,1.L.H /,Tr/ for which the analog of Theorem 9.7.5 continues to hold.

Part IV Applications to Noncommutative Geometry

This part offers a new and concise treatment of the noncommutative residue in Alain Connes’ quantum calculus. Some aspects concerning the use of Dixmier traces in noncommutative geometry that we discuss are not commonly known and have not appeared before in texts on noncommutative geometry. Introduced in the subsequent chapters is a new notion of the noncommutative residue and new ways to calculate the Dixmier trace using expectation values instead of singular values. By this approach we prove a fundamental trace theorem which provides Connes’ pivotal trace theorem as a corollary. Using this new approach we revisit isospectral deformations of manifolds, the most common examples of noncommutative geometries. We finish with the fascinating link between Dixmier traces and classical limits. An overview of this part is given below. The notes after the overview indicate some of the many applications of singular traces to noncommutative geometry outside of our subsequent discussion. Alain Connes’ noncommutative geometry concerns, generally, continuous traces on the Lorentz ideal M1,1 of compact operators on a separable Hilbert space. The use of traces in noncommutative geometry can also be phrased in terms of general traces on the weak-l1 ideal L1,1 of compact operators. Both of these ideals of compact operators were introduced in Chapter 1 and both are associated to the harmonic sequence. In Chapter 10 a summary of the theory of traces on the operator ideals L1,1 and M1,1 is given, using the results in the previous chapters. We consider only ideals of compact operators in subsequent chapters. The summary in Section 10.1 is a convenient reference on Dixmier traces on the Dixmier-Macaev ideal. Chapter 10 also introduces the calculus of pseudo-differential operators (integral operators that generalize differential operators). The calculus is operator based and it is the prototype for Connes’ quantum calculus in noncommutative geometry. In the pseudo-differential operator calculus, differentiation is observed in commutation by first order pseudo-differential operators and integration is observed by traces on pseudo-differential operators of negative order. The trace Tr on trace class operators can be applied to compactly supported pseudo- differential operators on the Euclidean space Rd that are of order strictly less than d d since they extend to trace class operators on the Hilbert space L2.R /.Mariusz Wodzicki [257] established the existence of a distinct trace ResW on the classical compactly supported pseudo-differential operators of order d, called the noncommutative residue. The noncommutative residue of Wodzicki is defined by integrating 304 Part IV Applications to Noncommutative Geometry the principal symbol of the classical pseudo-differential operator over the co-sphere bundle on Rd . The difference between the canonical trace and the noncommutative residue on classical operators is revealed by the singular properties of the residue. Connes’ Trace Theorem, proven by Alain Connes in [45], was the first result to equate Wodzicki’s residue with a singular trace. It states that a compactly supported 1 Rd 1 Rd classical pseudo-differential operator A : Cc . / ! Cc . / of order d extends to a compact linear operator belonging to L1,1 and

Tr!.A/ D ResW .A/,

1 where ResW .A/ is the noncommutative residue of A and Tr!.A/ is any Dixmier trace applied to the extension of A. The theorem indicates that the noncommutative residue of A is given by a singular trace on the weak-l1 ideal, and, conversely, since the noncommutative residue can be calculated from the symbol of A, it provides a tractable formula for calculating the Dixmier trace of A. Connes’ insight provides a stunning link between a construct of operator algebras on the left of the above display (a singular trace), and a construct of differential geometry on the right (the noncommutative residue). Connes’ Trace Theorem is usually where the foundation of the noncommutative residue in noncommutative geometry ends and applications begin (the rationale derived from the theorem is that Dixmier traces are the “noncommutative residue” for noncommutative versions of pseudo-differential operators of order d). There is, however, more to the story of Connes’ Trace Theorem. Chapter 11 derives the trace theorem originally stated by Connes as a special case of a more general theorem. This general theorem is the core of our application of singular traces, and utilizes the deep theory of commutator subspaces. To prove the more general theorem we introduce, in Chapter 11, Nigel Kalton’s notion of an operator modulated with respect to a positive bounded linear operator on a separable Hilbert space. Let denote the Laplacian on Rd .The.1 /d=2- d modulated operators, in Kalton’s sense, on the Hilbert space L2.R / we call the Lapla- cian modulated operators. According to the general theory described in Chapter 11, each Laplacian modulated operator is Hilbert–Schmidt. It is well known that each d d Hilbert–Schmidt operator on L2.R / is an integral operator with kernel K 2 L2.R d d d R / and symbol p 2 L2.R R /. If the kernel K is (almost everywhere) compactly supported, then the Hilbert–Schmidt operator is called compactly supported. We show in Chapter 11 that a compactly supported Laplacian modulated operator extends the notion of a compactly supported pseudo-differential operator of order d.Wein- troduce a vector-valued noncommutative residue on the set of compactly supported

1 We caution the reader already familiar with Connes’ theorem that, in deﬁning the scalar noncommutative residue, we scale the usual formula by the reciprocal of d. Part IV Applications to Noncommutative Geometry 305

Laplacian modulated operators by setting Z Z 1 1 Res.T / :D pT .u, s/duds (IV.1) =d d log.2 C n/ jsjn1 R nD0 where Œ denotes the equivalence class in l1=c0, T is a compactly supported Laplacian modulated operator, and pT is the symbol of T . In Chapter 11 we show that Res is an extension of the noncommutative residue of Wodzicki, ResW . Any extended limit ! on l1 (a state on l1 that vanishes on c0) deﬁnes a Dixmier trace Tr! on the weak-l1 ideal L1,1 (see Section 9.7). Extended limits lift immediately to functionals ! : l1=c0 ! C, such that !.Œa/ D Œa D lim a if a is a convergent sequence. In fact, extended limits are exactly the states on the algebra l1=c0.The following Trace Theorem is the main result of Part IV and is the connection between singular traces and the noncommutative residue.

Trace Theorem Every compactly supported Laplacian modulated operator T belongs to L1,1 and

(a) Tr!.T / D !.Res.T //

for any Dixmier trace Tr!.

(b) Tr!.T / D Res.T /

for all Dixmier traces Tr! if and only if Res.T / is scalar valued. (c) '.T / D Res.T / L 1 1 for every trace ' on 1,1 such that '.diagf.1 C n/ gnD0/ D 1 if and only if Z Z

pT .u, s/duds D Res.T / log n C O.1/, n 1. jsjn1=d Rd

Here diag is the diagonal operator for any chosen basis and Res is the vector-valued residue deﬁned in (IV.1). Chapter 11, in particular Sections 11.5 and 11.7, list the corollaries to the Trace Theorem. Besides having Connes’ Trace Theorem as a corollary, the theorem can be used to show that integration of functions can be recovered using a singular trace.

Integration of functions using singular traces d If f 2 L2.R / is compactly supported and Mf is the normal operator densely deﬁned 1 Rd d=2 L by .Mf x/.u/ D f .u/x.u/, x 2 Cc . /,thenMf .1 / 2 1,1 and Z Sd1 d=2 Vol. / '.Mf .1 / / D f.u/du (IV.2) d.2/d Rd L 1 1 for every trace ' on 1,1 with '.diagf.1 C n/ gnD0/ D 1. 306 Part IV Applications to Noncommutative Geometry

Another corollary of the Trace Theorem observed in Section 11.5 is that there are pseudo-differential operators that are not measurable operators in Connes’ sense. We can conclude that the general class of compactly supported pseudo-differential operators of order d does not have a unique trace; although the subclass of classical operators does have a unique trace (Wodzicki’s noncommutative residue). Given the machinery for compactly supported operators on Rd , it is a fairly routine task to transfer the Trace Theorem above to closed Riemannian manifolds. On a manifold the relevant operators are Hodge–Laplacian modulated operators which are shown to be locally Laplacian modulated. The passage to closed manifolds is routine, however, there is one critical implication.

Spectral formula of the scalar noncommutative residue If A is a classical pseudo-differential operator of order d onaclosedd-dimensional Riemannian manifold X, and ResW is Wodzicki’s noncommutative residue on X,then

1 Xn ResW .A/ D lim hAej , ej i, n!1 log.2 C n/ j D0

1 where fej gj D0 is an orthonormalbasis of the Hilbert space L2.X/ consisting of eigen- 2 2 2 vectors of the Laplace–Beltrami operator such that ej D j ej , 0 1 are increasing.

In fact, we deﬁne the noncommutative residue of a Hodge–Laplacian modulated operator T on a closed Riemannian manifold by 1 Xn 1 Res.T / :D hTe , e i log.2 C n/ j j j D0 nD0

1 where Œ denotes the equivalence class in l1=c0 and fej gj D0 is a basis of eigenvectors of the Laplace–Beltrami operator as above. The critical feature of the spectral formula for the noncommutative residue on a closed Riemannian manifold is that it can be extended to any unbounded Fredholm module in the noncommutative geometry of Connes.

Connes’ quantum calculus is based on the pair .H , D/ of a separable Hilbert space H and an unbounded self-adjoint operator D :Dom.D/ ! H , which is the analog of a Dirac operator [146]. The positive operator D2 is the analog of a Laplacian. The functional calculus of positive operators allows the deﬁnition of the positive operator with trivial kernel: hDir :D .1 C D2/r=2, r 2 R. Part IV Applications to Noncommutative Geometry 307

The pair .H , D/ is said to have dimension d>0ifD has compact resolvent and

d hDi 2 L1,1.

In Chapter 12 we deﬁne, in analogy to the noncommutative residue on closed manifolds, the vector-valued noncommutative residue 1 Xn 1 Res .T / :D hTe , e i , (IV.3) D log.2 C n/ j j j D0 nD0

d where Œ denotes the equivalence class in l1=c0, T 2 L.H / is a hDi -modulated 1 2 2 operator, and fej gj D0 is an orthonormal basis of eigenvectors of D with D ej D 2 2 2 j ej ,where0 1 are increasing. The main result on modulated operators in Chapter 11, Theorem 11.2.3, is central to the residue ResD being well deﬁned and it allows noncommutative trace theorems, saying that Dixmier traces can be calculated using expectation values.

Trace Theorem (NCG) d Every hDi -modulated operator T belongs to L1,1 and

(a) Tr!.T / D !.Res.T //

for any Dixmier trace Tr!.

(b) Tr!.T / D Res.T /

for all Dixmier traces Tr! if and only if Res.T / is scalar valued. (c) '.T / D Res.T / L 1 1 for every trace ' on 1,1 such that '.diagf.1 C n/ gnD0/ D 1 if and only if Xn hTej , ej iDRes.T / log n C O.1/, n 1. j D0

The Trace Theorem in noncommutative geometry, where the spectral characteris- tics are predominant, has the same outcome as the Trace Theorem on manifolds: com- putability. Knowing the eigenvectors and eigenvalues of D allows explicit calculation of the traces of modulated operators and explicit conditions for measurability. This is seen especially for integration in Connes’ quantum calculus. The functional

d Int! : S ! Tr!.ShDi /, S 2 L.H /, (IV.4) where ! is an extended limit on l1, is considered to be the integral in noncommutative geometry. It is an evident extension of the formula (IV.2). All bounded operators 308 Part IV Applications to Noncommutative Geometry

S 2 L.H / have the property that ShDid is a hDid -modulated operator. Using the Trace Theorem, and assuming a Weyl asymptotic condition on the eigenvalues of D, 1=d jnjn , n 0, we show in Chapter 12 that computing the integral in (IV.4) is equivalent to computing the logarithmic mean of the expectation values of S: 1 L Int!.S/ D ! ı M fhSen, enignD0 , S 2 .H /, where P n 1 xj Pj D0 1Cj 1 Mn.x/ :D n 1 , x Dfxj gj D0 2 l1, j D0 1Cj

1 and M.x/ DfMn.x/gnD0. In particular, Int!.S/ is independent of ! if and only if the logarithmic means of the expectation values of S converge. Linking the logarithmic mean realization of the noncommutative integral to the heat kernel formulas of Chapter 8, we obtain, for S 2 L.H /, ˇ 1D2 1 Tr.Se / Int!.S/ D ! ı M fhSen, enig D D ! ı M . (IV.5) n 0 Tr.eˇ 1D2 /

Here ! is an extended limit on L1.0, 1/ such that ! and !ılog are dilation invariant, and the two formulas on the left involve the restriction of ! to an extended limit on l1. Equation (IV.5) represents, for d-dimensional unbounded Fredholm modules, a mathematical link between singular traces, the correspondence principle in quantum mechanics, and high temperature limits. It is explained further in Section 12.6. It is perhaps surprising in itself that there are interesting examples of the noncommutative residue (IV.3), and the integral (IV.4), outside of the pseudo-differential operator calculus. After all, (IV.3) and (IV.4) are, on the surface, just a residue and an integral by analogy. That the range of examples has been so astonishing, and outside the standard differential calculus, from fractional dimensions, to foliations, to isospectral noncommutative deformations, has been the driver of the growth of Connes’ noncommutative geometry. Using the noncommutative residue approach we look at the speciﬁc example of isospectral deformations in Chapter 12. The main result is that the noncommutative residue is invariant under isospectral deformation.

Notes

Our introduction to the applications has made various claims without detailed reference to the text. The reader can ﬁnd the location of these facts in the subsequent chapters as follows. Pseudo-differential operators on Rd , the calculus of pseudo-differential operators, and the regularity of pseudo-differential operators, are established in Section 10.2. Classical pseudo- differential operators on Rd , principal symbols and the noncommutative residue are discussed Part IV Applications to Noncommutative Geometry 309

in Section 10.2; in particular the trace-like and singular-like properties of the noncommutative residue. d Section 11.3 discusses Laplacian modulatedoperators on L2.R / and the properties of their symbols. In this section can be found the result that all compactly based pseudo-differential operators of order d are Laplacian modulated, Proposition 11.3.17. The residue in (IV.1) is shown to be well defined, Proposition 11.3.18, and be an extension of the noncommutative residue, Proposition 11.3.21. An example of a compactly based pseudo-differential operator of order d whose residue is the class of the oscillating sequence sin log log n1=d is given in Proposition 11.3.22. Section 11.5 contains the general trace theorem, Theorem 11.5.1. Connes’ Trace Theorem is Corollary 11.5.4. The existence of a non-classical pseudo-differential operator which is not Dixmier measurable is shown in Corollary 11.5.3. Hodge–Laplacian modulated operators on closed Riemannian manifolds are introduced in Section 11.6. Their equivalencelocally to compactlysupported Laplacianmodulated operators is shown in Proposition 11.6.7. Definition 11.6.6 introduces the residue of Hodge–Laplacian modulated operators, which is shown to be well defined by using Theorem 11.2.3. In many regards the closed Riemannian manifold example is simpler than the space Rd . The residue of so-called localized Hodge–Laplacian modulated operators is shown to be locally equivalent to the Rd residue, and from this follows the spectral formula for Wodzicki’s noncommutative residue, Corollary 11.6.16. That the noncommutative residue ResD is well defined for any unbounded Fredholm module of finite dimension is shown in Lemma 12.2.4. The trace theorem for unbounded Fredholm modules is Theorem 12.2.6. Theorem 12.3.1 provides the formula equating the noncommutative integral with the logarithmic means. Theorem 12.6.1 combines the logarithmic mean formula with the heat kernel formula to relate the noncommutative integral to the high temperature Gibbs states. The result that the noncommutative residue is an invariant of isospectral deformations can be found in Theorem 12.4.1.

Applications Outside the Content of this Book Dixmier traces feature in Connes’ interpretation of noncommutative residues, and there are many originating examples in Connes’ monograph [54]. Specifically there are applications to mathematical physics, Einstein–Hilbert actions and particle physics’ standard model [36, 50, 131], applications in string theory and Yang–Mills theory [45, 51, 75, 218], fractals [48, 104], isospectral deformations [53,96], foliations and noncommutative index theory [9,30–32,54], and Toeplitz and Hankel operators on the unit disc [84, 85]. See [189] for initial work on non-compact manifolds without boundary, [169] for consideration of compact manifolds with boundary, and [95] for a discussion of Moyal planes (non-commuting Euclidean coordinates). For a general consideration of the noncommutative integral for a semifinite unbounded Fredholm module .H , D/,whereD does not have compact resolvent, see [28]. For applications of Dixmier traces in general semifinite von Neumann algebras see [9,32,177].

Chapter 10 Preliminaries to the Applications

10.1 Summary of Traces on L1,1 and M1,1

In this section we collect facts from Chapters 1–9 about traces on the two-sided ideals of compact operators L1,1 and M1,1. Suppose H is a separable Hilbert space and .A/ is the singular value sequence of an operator A 2 L.H /. Alain Connes’ noncommutative geometry generally concerns traces on the symmetric ideal of compact operators

M1,1 :DfA 2 L.H / : .A/ 2 m1,1g, corresponding to the Sargent sequence space

1 Xn m1,1 :Dfa 2 l1 : kakm1,1 :D sup .k, a/ < 1g. n0 log.2 C n/ kD0

A symmetric norm on M1,1 is given by

M kAkM1,1 :Dk.A/km1,1 , A 2 1,1.

Since l1 m1,1 (where l1 denotes summable sequences) then the trace class opera- L M L tors 1 form a symmetric sub-ideal of 1,1. The closure of 1 in the norm kkM1,1, denoted .M1,1/0, is a proper sub-ideal M1,1.Theideal.M1,1/0 is called the separable part of M1,1. Note, for instance, that the separable part does not contain the 1 1 operator diag.f.1 C n/ gnD0/ (see Lemma 1.2.8). Here, diag denotes the diagonal operator for an arbitrary orthonormal basis of H .

Facts about Continuous Traces M M A continuous trace on 1,1 is a linear functional ' 2 1,1 such that

'.ŒA, B/ D 0, A 2 M1,1, B 2 L.H /.

A continuous trace ' is normalized if

1 1 .' ı diag/.f.1 C n/ gnD0/ D 1, 312 Chapter 10 Preliminaries to the Applications

and is fully symmetric if, given 0 A, B 2 M1,1,

'.A/ '.B/ when A B.

Here B A denotes the Hardy–Littlewood submajorization

Xn Xn .k, B/ .k, A/, n 0, kD0 kD0 explained in Section 3.3. Theorem 10.1.1 collates results from prior chapters about continuous traces and fully symmetric traces. Let .A/ denote an eigenvalue sequence of a compact operator A, that is, the non-zero eigenvalues of A ordered, with multiplicity, such that j.n, A/j, n 0, are decreasing (we append zeros if A has only ﬁnitely many non-zero eigenvalues, and take the zero sequence as the eigenvalue sequence if A has no non-zero eigenvalues). In the following theorem ' denotes a continuous trace on M1,1.

Theorem 10.1.1 (Continuous traces on M1,1).

(a) All continuous traces on M1,1 are spectral (depend only on eigenvalues),

'.A/ D .' ı diag/..A//, A 2 M1,1,

where .A/ 2 m1,1 is an eigenvalue sequence of A.

(b) All continuous traces on M1,1 are singular (vanish on trace class operators),

'.A/ D 0, A 2 L1.

(d) There are normalized positive continuous traces on M1,1 other than Dixmier traces.

(e) There is no unique normalized continuous trace on M1,1. In fact, the Hermitian continuous traces form an inﬁnite dimensional Banach lattice.

Proof. The assertion of (a) is proved in Corollary 5.6.4. Every positive trace on M1,1 necessarily vanishes on L1. The assertion of (b) now follows from Theorem 4.2.2. The assertion of (c) is proved in Theorem 6.4.1. The assertion of (d) is proved in Theorem 4.8.1. The assertion of (e) is proved in Theorem 4.3.5 and Theorem 4.10.1. Section 10.1 Summary of Traces on L1,1 and M1,1 313

Facts about Dixmier Traces

The set of Dixmier traces on M1,1 is exactly the set of all normalized fully symmetric traces on M1,1. As explained in Chapter 1 and Chapter 6, Dixmier traces have the explicit formulation 1 Xn 1 Tr!.A/ :D ! .k, A/ ,0 A 2 M1,1, (10.1) log.2 C n/ D kD0 n 0 where ! is a dilation invariant extended limit on l1. The formula Tr! is extended to non-positive operators linearly. We can equivalently consider ! a dilation invariant extended limit on L1.0, 1/, see Chapter 6, in which case we understand the sequence of singular values is treated as a step function and Z 1 t !.A/ :D ! .s, A/ds ,0 A 2 M1,1, (10.2) log.1 C t/ 0 where ! is a dilation invariant extended limit on L1.0, 1/. We continue to denote the discrete and continuous version of the Dixmier trace separately, by Tr! and ! respectively. The most convenient way to translate formulas for the continuous version of the Dixmier trace to the discrete version is to restrict an extended limit ! on L1.0, 1/ to 1 the step functions on the partition f.n, n C 1gnD0. These step functions are in isometric correspondence with the sequence space l1, and the restriction of ! to such step functions is an extended limit on l1. A convenient way to notate this restriction is to use the expectation operator E deﬁned in Deﬁnition 3.6.1, and then, for every dilation invariant extended limit on L1.0, 1/ we have (see Theorem 6.4.4):

Tr!ıE D !.

Either version of the Dixmier trace, discrete or continuous, has its advantages. Alain Connes called an operator A 2 M1,1 measurable if Tr!.A/ is the same value for all dilation invariant extended limits ! (which is equivalent to !.A/ having the same value for all such !, see Theorem 6.4.1 and Theorem 6.4.4). We considered various forms of measurability in Chapter 9, and refer to Tr!.A/ having the same value for all dilation invariant extended limits ! on l1 by the terminology that the operator A is Dixmier measurable, or that A is measurable in the sense of Connes. Calculation of a Dixmier trace can be difﬁcult since eigenvalues and singular values can be intractable. Chapters 7 and 8 looked at various formulations of the Dixmier trace that have been developed for the purpose of improved calculation; and to develop the analogy between the Dixmier trace of a positive operator in Connes’ quantum calculus and the -function residue and heat kernel asymptotic formulas for elliptic pseudo- differential operators. 314 Chapter 10 Preliminaries to the Applications

An operator A 2 M1,1 is called Tauberian if

1 Xn lim .k, A/ n!1 log.2 C n/ kD0 exists, where .A/ 2 m1,1 is an eigenvalue sequence of A. A positive compact operator 0 A 2 M1,1 with singular values

n n2 n .A/ D sup 2 Œ0,22 /, (10.3) n0 is an example of an operator which is not Tauberian (see Example 9.7.6). If 0 A 2 M1,1 and if ! is an extended limit on L1.0, 1/, the formula X1 1 1 1 1 .A/ :D ! Tr.A1C s / D ! .k, A/1C s ! s s kD0 is a called a -function residue. Similarly, 1 1 .A/ :D .! ı M/ Tr.e.tA/ / ! t is called a heat kernel formula where Z 1 t ds .M x/.t/ :D x.s/ , x 2 S.0, 1/. log.t/ 0 s Collated results from prior chapters concerning these formulas for the Dixmier trace are presented in Theorem 10.1.2. In the theorem the notation ! denotes a Dixmier trace as deﬁned in (10.2).

Theorem 10.1.2 (Formulas for Dixmier traces). (a) Every Dixmier trace has an explicit spectral formula, 1 Xn Tr .A/ D ! .k, A/ , A 2 M 1, ! log.2 C n/ 1, kD0

where .A/ 2 m1,1 is an eigenvalue sequence of A.Here! is a dilation invariant extended limit on l1.

(b) If A 2 M1,1 is Tauberian, then A is Dixmier measurable. The converse statement is true if A 0 but is false in general. Section 10.1 Summary of Traces on L1,1 and M1,1 315

(c) If the extended limit ! on L1.0, 1/ is dilation invariant and ! ı log is dilation invariant, then ! D !ılog. However, there exists a dilation invariant extended limit ! on L1.0, 1/ such that ! 6D !ılog. (d) The set of -function residues is strictly smaller than the set of Dixmier traces.

(e) If the extended limit ! on L1.0, 1/ is M -invariant (! D !ıM ), then ! D !. However, there exists a dilation invariant extended limit ! such that ! ¤ ! . (f) Yet, the set of heat kernel formulas is the same as the set of Dixmier traces.

(g) If A 0 is Dixmier measurable, then 1Cs 1 .A/1 !.A/ D lim sTr.A / D lim M Tr.e / , s!0 !1

for every extended limit ! on L1.0, 1/. Conversely, the existence of the latter limits guarantee that the operator A 0 is Dixmier measurable.

Proof. The assertion of (a) is proved in Theorem 7.3.1. The ﬁrst assertion of (b) follows from (a). The second assertion of (b) is proved in Theorem 9.2.1 and Exam- ple 9.7.6. The assertion of (c) is proved in Theorem 8.6.8. The assertion of (d) is proved in Theorem 8.7.1. The assertion of (e) is proved in Theorem 8.2.9. The assertion of (f) is a combination of Theorem 8.3.6 and Theorem 6.4.1. The assertion of (g) is proved in Theorem 9.3.1.

Facts about Traces on the Ideal L1,1 Most applications in noncommutative geometry involve compact operators with sin- 1 gular values O..1Cn/ /, n 0. The set of such compact operators form the weak-l1 ideal 1 L1,1 :DfA 2 L.H / : .A/ D O..n C 1/ /g, which, through the Calkin correspondence, corresponds to the weak-l1 sequence space l1,1. The ideal has a quasi-norm given by

kAk1,w :D sup.1 C n/.n, A/, A 2 L1,1. n0

A trace on L1,1 is a linear functional ' : L1,1 ! C such that

'.ŒA, B/ D 0, A 2 L1,1, B 2 L.H /.

Because L1,1 is a quasi-normed, but not symmetrically normed, ideal, we do not iso- late, in our consideration, (quasi-norm) continuous traces on L1,1 from other unitarily 316 Chapter 10 Preliminaries to the Applications

invariant linear functionals. The traces we are considering on L1,1 are not assumed to have any continuity or positivity properties. A trace ' on L1,1 is normalized if

1 1 .' ı diag/.f.1 C n/ gnD0/ D 1.

We note that a Dixmier trace on M1,1 restricts to a normalized trace on L1,1,and that the restriction is continuous for the quasi-norm on L1,1. Further, for any extended limit ! on l1, the formula Tr! on L1,1 deﬁned by (10.1) is the restriction of some Dixmier trace on M1,1 (see Section 9.7). In the following theorem ' denotes a trace on L1,1,and.M1,1/0 denotes the separable part of M1,1.

Theorem 10.1.3.

(a) All traces on L1,1 are spectral,

'.A/ D .' ı diag/..A//, A 2 L1,1,

where .A/ 2 l1,1 is an eigenvalue sequence of A.

(b) All traces on L1,1 are singular (in fact, vanish on trace class operators),

'.A/ D 0, A 2 L1.

for all traces ' on L1,1 if and only if A 2 L1.

(d) If A 2 L1,1, then

Tr!.jAj/ D 0

for all Dixmier traces Tr! restricted to L1,1 if and only if A 2 L1,1 \.M1,1/0.

(e) There are normalized traces on L1,1 which are not the restriction to L1,1 of any Dixmier trace on M1,1.

(f) A 2 L1,1 is Dixmier measurable if and only if it is Tauberian (i.e. there is a constant c such that

1 Xn .k, T/D c C o.1/, n 0 log.2 C n/ kD0

where .A/ 2 l1,1 is an eigenvalue sequence of A). Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 317

(g) If A 2 L1,1 and ' is a normalized trace on L1,1,then

'.A/ D c

for a constant c independent of ' if and only if 1 Xn 1 .k, T/D c C O , n 0, log.2 C n/ log.2 C n/ kD0

where .A/ 2 l1,1 is an eigenvalue sequence of A.

Proof. The assertion of (a) is proved in Theorem 5.7.6. The assertion of (b) is proved in Corollary 5.7.7. The assertions of (g) and of (c) is proved in Theorem 5.7.6. Ex- plicitly, to see (g), '.A/ D c for all normalized traces if and only if A cdiag.f.1 C 1 1 L 1 1 n/ gnD0/ 2 Com. 1,1/.P Theorem 5.7.6 proves that A cdiag.f.1 C n/ gnD0/ 2 L n Com. 1,1/ if and only if j D0 .j , A/ c log.2 C n/ D O.1/. The assertion of (f) is proved in Theorem 9.7.5. The assertion of (d) follows from that (f) (or Theo- rem 9.2.1). It follows from (d) and (c) that there exists a positive operator A 2 L1,1, A 62 L1 which nulliﬁes all Dixmier traces, but not all traces. Hence, the set of all traces is strictly larger then the set of all Dixmier traces. This proves the assertion of (e).

10.2 Pseudo-differential Operators and the Noncommutative Residue

The purpose of this section is to introduce the noncommutativeresidue and the pseudo- differential calculus of pseudo-differential operators on Rd , and to understand the origin of Connes’ noncommutative calculus. Standard results in pseudo-differential operator theory are given without proof. Detailed references are indicated in the section end notes.

Pseudo-differential Operators on Rd

Let Rd denote the Euclidean space of dimension d. A multi-index of order jˇj is

Zd ˇ D .ˇ1, :::, ˇd / 2 C such that

jˇj :D ˇ1 CCˇd . 318 Chapter 10 Preliminaries to the Applications

For partial derivatives we use the notation

jˇ j ˇ @ Rd @s :D , s D .s1, :::, sd / 2 . ˇ1 ˇd @s1 , :::, @sd

The inner productp on Rd we denote by hs, ui, s, u 2 Rd , and the associated vector Rd 1 Rd S Rd norm jsjD hs, si, s 2 . The notation Cc . / and . / denote the spaces of smooth functions of compact support and Schwartz functions (the smooth functions of rapid decrease), respectively. The latter is a Fréchet space for the family of seminorms

˛ ˇ p˛,ˇ ./ :D sup jjs j.@s /.s/j s2Rd where ˛, ˇ runs over all multi-indices. The space of distributions, D0.Rd /, is the topo- 1 Rd S0 Rd logical dual of Cc . /, and the space of tempered distributions, . /, is the topological dual of the Fréchet space S.Rd /. 1 Rd If 2 Cc . / is positive, Z .s/ds D 1 Rd and Z d 1 1 Rd lim . s/x.u s/ds D x.u/, u 2 Cc . / !0C Rd

d 1 then the family f.s/ :D . s/g>0 is called a molliﬁer or an approximate identity. In the sense of distributions, lim!0C D ı where ı denotes the Dirac distribution. Let

r :D .@s1 , :::, @sd / denote the gradient r,and

:D.@2 CC@2 / s1 sd denote the Laplacian . Finally, we use the standard notation

hsi :D .1 Cjsj2/1=2, s 2 Rd , which may also apply to operators, for example

hiri2k :D .1 /k, k 2 N. Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 319

Deﬁnition 10.2.1. If a 2 C 1.Rd Rd Rd / and there is a ﬁxed value m 2 R such that, for each multi-index ˛, ˇ,

˛ ˇ mjˇ j Rd [email protected],v/@s a.u, v, s/jDO.hsi /, u, v, s 2 , (10.4) we say that a is an amplitude of order m.

Remark 10.2.2. In general terms we have introduced the uniform amplitudes of Hörmander type (1,0), see the end notes. Note that the right-hand side of (10.4) is independent of u and v. The following are standard results, see e.g. [201, Chapter 2] and the end notes. We omit the proofs.

Proposition 10.2.3. If a is an amplitude of order m, then the operator Z Z .A/.u/ :D e2ihuv,sia.u, v, s/.v/dvds, 2 S.Rd / (10.5) Rd Rd is a continuous linear operator

A : S.Rd / ! S.Rd /.

Deﬁnition 10.2.4. The integral operator A in (10.5) is called a pseudo-differential operator of order m 2 R. The set of all pseudo-differential operators of order m we denote by Lm. If a is an amplitude associated to a pseudo-differential operator A 2 Lm,deﬁne Z Z Z hA , i :D e2ihuv,sia.u, v, s/ .u/.v/dvdsdu, , 2 S.Rd /. Rd Rd Rd

The adjoint A is deﬁned by

hA , i :D hA, i, , 2 S.Rd /, and A is a pseudo-differential of order m associated to the amplitude

a.u, v, s/ :D a.v, u, s/, u, v, s 2 Rd .

The adjoint is the natural way to extend pseudo-differential operators to linear operators on tempered distributions.

Proposition 10.2.5. An operator A 2 Lm has a continuous extension

A : S0.Rd / ! S0.Rd / 320 Chapter 10 Preliminaries to the Applications given by .A /./ :D .A/, 2 S0.Rd /, 2 S.Rd /.

0 d d m The kernel KA 2 S .R R / of a pseudo-differential operator A 2 L is the tempered distribution densely deﬁned by

d KA . ˝ / :DhA , i, , 2 S.R /.

Deﬁnition 10.2.6. If p 2 C 1.Rd Rd / and there is a ﬁxed value m 2 R such that, for each multi-index ˛, ˇ,

˛ ˇ mjˇ j Rd j@u@s p.u, s/jDO.hsi /, u, s 2 , (10.6) we say that p is a symbol of order m. If needed, the linear space of symbols of order m is denoted by Sm. Every pseudo-differential operator associated to a uniform amplitude has a unique 2ihu,si d symbol. Let es.u/ :D e , s 2 R .

Theorem 10.2.7. A continuous linear operator A : S0.Rd / ! S0.Rd / is a pseudo- differential operator of order m if and only if

d p.u, s/ :D es.u/.Aes /.u/, u, s 2 R , (10.7) is a symbol of order m. From (10.7), p is the unique symbol such that Z .A/.u/ D e2ihu,sip.u, s/.F/.s/ds, 2 S.Rd /, (10.8) Rd where F denotes the Fourier transform Z .F/.s/ :D e2ihs,vi.v/dv, s 2 Rd , 2 S.Rd /. Rd

This provides a one-to-one correspondence between pseudo-differential operators of order m and symbols of order m. The function p in (10.7) or (10.8) is called the symbol of the pseudo-differential operator A. m If pA is the symbol of A 2 L , then the kernel KA has the form Z 2ihuv,si d KA.u, v/ D e pA.u, s/ds, u, v 2 R . (10.9) Rd

Generally (10.9) does not converge absolutely and the expression needs to be understood as an oscillatory integral [221, §2]. We avoid many of these technicalities since Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 321

d d the majority of operators we consider have symbols p 2 L2.R R /, removing the need for oscillatory integrals.

Deﬁnition 10.2.8. If p and q are symbols of order m, we write p q if p q is a symbol of order m 1. This is an equivalence relation on symbols of order m. If p is the symbol of a pseudo-differential operator A 2 Lm the equivalence class Œp 2 S m=Sm1 is called the principal symbol of A.

The pseudo-differential operators that are order m, for every m 2 R, are called smoothing operators (their kernels are smooth functions) and the set of smoothing operators we denote by L1. Denote the set of symbols of smoothing operators by S1.

Example 10.2.9. The fundamental pseudo-differential operator of order m deﬁned by, Z .1 /m=2.u/ :D e2ihu,sih2sim.F/.s/ds, 2 S.Rd /, (10.10) Rd extends the linear differential operator .1/k=2, k 2 2N. The symbol of .1/m=2 is h2sim.Sincehsim Djsjm.1 Cjsj2 /m=2, jsj > 1, by using the binomial theorem the principal symbol of .1 /m=2 consists of symbols that agree with .2/mjsjm, jsj > 1,uptoasymbolinS m1. Actually this agreement is only required outside an arbitrary neighborhood of 0. 1 Let g 2 C .RC/ be positive and increasing with 0 t<1=2 g.t/ :D , t>0. 1 t>1

Then .g.˛1jsj/ g.ˇ1jsj//jsjm 2 S 1, ˛, ˇ>0. Hence, we could equally say the principal symbol of .1/m=2 consists of symbols that agree with .2/mjsjm, jsj >, >0, up to a symbol in S m1.

In the previous example the principal symbol could be chosen as a homogeneous function when jsj1. We give an example of a pseudo-differential operator with a non-homogeneous principal symbol. This shall relate later to the existence of a non- classical pseudo-differential operator that is not measurable in the sense of Connes.

Example 10.2.10. A smooth function q on Rd such that

q.s/ :Djsjm .sin C cos/.log.log.jsj///, jsj > 4, is a symbol of order m, and there is a pseudo-differential operator Q 2 Lm (with symbol q) whose principal symbol is not homogeneous for jsj > 4. 322 Chapter 10 Preliminaries to the Applications

Proof. The inequality

˛ j˛j mj˛j j@s q.s/jC .d Cj˛j/Šjsj , jsj > 4 is straightforward where C>0 is a constant. Thus, q 2 Sm is a symbol and deﬁnes a unique pseudo-differential operator Q of order m. Note that log.2/ log.log.2jsj// D log.log.jsj// C log 1 C log.jsj/ log.2/ 1 D log.log.jsj// C C O log.jsj/ log2.jsj/ and, for t>0,

.cos C sin/.t C ı/ D .cos C sin/.t/ C .cos sin/.t/ı C O.ı2/.

Hence, log.2/ jsjm q.2jsj/ 2mq.jsj/ D 2m jsjm .cos sin/.log.log jsj// C O . log.jsj/ log2.jsj/

It follows that q.2jsj/ 2mq.jsj/ ¤ O.jsjm1 /.

Hence, the principal symbol of Q cannot be homogeneous.

Calculus and regularity of pseudo-differential operators The relationship between symbols and operators is unique, but that between amplitudes and operators is not. The following formula relates amplitudes and symbols. 1 mj Let fpj gj D0 such that pj 2 S , m0 >m1 > , mj !1as j !1.If p 2 S m0 then by an asymptotic expansion or sum, X p pj , j we mean that Xn mn p pj 2 S , n 0. j D0 Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 323

Proposition 10.2.11. If P 2 Lm with amplitude a and symbol p then

X ˛ .i/ ˛ ˛ p.u, s/ .@ @ a.u, v, s//j D ˛Š s v v u j˛j where the asymptotic sum runs over all multi-indices ˛. The above formula is central to showing that the pseudo-differential operators associated to the uniform symbols form a graded algebra with adjoint, called the pseudo- differential calculus.

Proposition 10.2.12. Let Lm be the pseudo-differential operators of order m. (a) If A 2 Lm,thenA 2 Lm and X .i/˛ p .u, s/ .@˛@˛p /.u, s/. A ˛Š s u A j˛j

(b) If A 2 Lm1 and B 2 Lm2 ,thenAB 2 Lm1Cm2 and X .i/˛ p .u, s/ .@˛p /.u, s/.@˛p /.u, s/. AB ˛Š s A u B j˛j

Example 10.2.13 (Multiplication operators). Asymptotic formulas are a convenient 1 Rd way to show the following results. Let 2 Cb . /, which denotes the smooth functions such that ˛ sup j.@s /.s/jDO.1/ s2Rd

0 for each multi-index ˛.LetM 2 L be the zero order pseudo-differential operator of multiplication by the function ,

d .M /.s/ :D .s/ .s/, 2 S.R /.

1 Rd m=2 (a) Let , 2 Cb . / such that D . By Proposition 10.2.12 M.1/ 2 m m=2 m L and M .1 / M 2 L . It is readily checked that

m m= pM .1/ 2 .u, s/ D .u/h2si .

m m=2 Since a.u, v, s/ D .u/h2si .v/ is an amplitude for M.1 / M ˛ then, using Proposition 10.2.11 (.@v /.v/jvDu D 0 when u 2 supp./ and ˛ supp@u.u/ supp..u//),

m 1 m= pM .1/ 2M .u, s/ .u/h2si 2 S . 324 Chapter 10 Preliminaries to the Applications

Hence, m=2 m=2 1 M .1 / M.1 / M 2 L .

1 Rd (b) Let , 2 Cb . / such that D 0. By Proposition 10.2.11 we have that

1 m= pM .1/ 2M 2 S .

Hence, m=2 1 M .1 / M 2 L .

1 Rd (c) Let 2 Cb . /. By Proposition 10.2.12 we have that

m1 m= m= pM .1/ 2 .u, s/ p.1/ 2M .u, s/ 2 S .

Hence, if ŒA, B :D AB BA denotes the commutator,

m=2 m1 Œ.1 / , M 2 L .

In fact, if A 2 Lm1 and B 2 Lm2 , generally by Proposition 10.2.12(b),

ŒA, B 2 Lm1Cm21,

so that the operators AB and BA have the same principal symbol.

A detailed view of the regularity of pseudo-differential operators arises from the consideration of Sobolev spaces. The Sobolev space H r .Rd /, r 2 R, consists of the functions Z 1=2 r d d 2r 2 H .R / :D f 2 L2.R / : kf kH r :D hsi j.Ff /.s/j ds < 1 , Rd

r d and it is a Hilbert space with the norm kf kH r , f 2 H .R /. The space of Schwartz functions, S.Rd /, is dense in the Sobolev space H r .Rd /,foranyr 2 R.Thereare r d r d evident continuous inclusions H 1 .R / H 2 .R / for r1 r2.

Theorem 10.2.14. Let A 2 Lm, m 2 R, be a pseudo-differential operator. Then A has an extension to a continuous linear operator

A : H r .Rd / ! H rm.Rd /, r 2 R.

From this fundamental theorem we gain important links to the functional analysis d of the Hilbert space L2.R /. Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 325

Proposition 10.2.15. Let Lm, m 2 R, be the set of pseudo-differential operators of order m. m m d d (a) If A 2 L , m>0, then the extension A : H .R / ! L2.R / is a closed linear operator.

0 d d (b) If A 2 L ,thenA has a bounded extension A : L2.R / ! L2.R /.

0 d d Proof. (b) Observed from Theorem 10.2.14 since H .R / D L2.R /.(a)From d d d d Proposition 10.2.3 we have that A : S.R / ! S.R /,henceS.R / L2.R / provides an invariant dense domain for A. The Hilbert space adjoint AQ has A AQ where A : S.Rd / ! S.Rd / is also densely deﬁned, hence A is closable. Since, using (b), there exists a constant C>0suchthatkkA :Dkk2 CkAk2 C kkH m, 2 S.Rd /,thenA is closed.

Example 10.2.16. The operator .1 /m=2, m<0, has a bounded extension (which we denote by the same symbol)

m=2 d d .1 / : L2.R / ! L2.R /, m<0.

m=2 m=4 2 Rd Since h.1/ u, uiDk.1/ uk2 > 0 for u 6D 0 2 L2. /, then the bounded operator .1 /m=2 is a positive operator. The extension .1 /m=2 : H m.Rd / ! d L2.R /, m>0, is an isometry of Hilbert spaces. This fact can be used to show (or even more simply that .1 /m=2 is unitarily equivalent to the multiplication operator h2sim via the Fourier transform) that the operator (again denoted by the same symbol) m=2 m d d .1 / : H .R / ! L2.R /, m>0 is an unbounded positive operator.

Using Example 10.2.16 we can equivalently write the Sobolev spaces H r .Rd /, r 2 R,as r d d r=2 H .R / :Dff 2 L2.R / : k.1 / f k2 < 1g, and with equivalent Hilbert space norm

r=2 kf kH r :Dk.1 / f k2.

Compactly Supported Pseudo-differential Operators

Pseudo-differential operators on Rd associated to the uniform class of symbols and of d d order m<0 generally do not extend to compact linear operators L2.R / ! L2.R /. Compactly based and compactly supported pseudo-differential operators provide a 326 Chapter 10 Preliminaries to the Applications sufﬁciently reduced class so that operators of order m<0 extend to compact linear operators and those of order m<d extend to trace class operators.

Deﬁnition 10.2.17. A pseudo-differential operator A : S.Rd / ! S.Rd / is said to be 1 Rd (a) compactly supported if there exist , 2 Cc . / such that MAM D A. 1 Rd (b) compactly based if there exists 2 Cc . / such that M A D A.

Here, M is a multiplication operator in the sense of Example 10.2.13. If K 2 S0.Rd Rd /, then denote by supp.K/ :D inffL Rd Rd jK. ˝ / D 0, , 2 S.Rd /, supp. ˝ / 2 Lcg the support of the distribution K and

Rd S Rd c supp1.K/ :D inffL jK. ˝ / D 0, , 2 . /, supp./ 2 L g the support in the ﬁrst variable of the distribution. The proof of the following result is omitted.

Proposition 10.2.18. Let A be a pseudo-differential operator.

(a) A is compactly supported if and only if its kernel KA has compact support. Equiv- alently, it has an amplitude a which is compactly supported in its ﬁrst 2 variables.

(b) A is compactly based if and only if its kernel KA has compact support in the ﬁrst variable. Equivalently, its symbol p is compactly supported in its ﬁrst variable. The compactly supported pseudo-differential operators form an algebra with adjoint, but the adjoint of a compactly based operator which is not compactly supported is not compactly based. The difference between compactly based and compactly supported operators is of order 1.

Proposition 10.2.19. Let A, B, C be pseudo-differential operators. (a) If A and B are compactly supported, then so are A, AB, BA.

(b) A is compactly supported if and only if both A and A are compactly based.

(d) If A is compactly based, then there exists a compactly supported operator A0 so that A A0 is of order 1.

Proof. Only the last property has to be proved. Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 327

1 Rd 0 Let MA D A for some 2 Cc . /.SetA D AM D MAM. Then A is compactly supported. Moreover, A0 has an amplitude p.u, s/.v/.Since .n/ 1 .v/jvDu D 0, when v 2 supp./,thenpA pA0 2 S by Proposition 10.2.11. Hence A A0 2 L1.

Let m, m 2 R, denote the set of functions p 2 C 1.Rd Rd / such that, for each multi-index ˛, ˇ,

˛ ˇ mjˇ jj˛j Rd j@u@s p.u, s/jDO.h.u, s/i /, u, s 2 . (10.11)

A pseudo-differential operator with symbol p 2 m is called a Shubin pseudo-differential operator of order m, see [221, IV]. The set of Shubin pseudo-differential m 1 m operators of order m is denoted by G . Denote G :D\mG . The Shubin pseudo- differential operators map the set of Schwartz functions to itself. They form a graded algebra with adjoint and have the following regularity theorem.

Theorem 10.2.20. Let A be a Shubin pseudo-differential operator of order m.Ifm< d d 0 then it has an extension A : L2.R / ! L2.R / that is a compact linear operator. If m<d then the extension A is Hilbert–Schmidt. If m<2d then the extension A is trace class.

A consequence is that the difference between compactly based and compactly supported pseudo-differential operators is trace class, for any order m.

Corollary 10.2.21. Let A be a compactly based pseudo-differential operator of order m, then there is a compactly supported pseudo-differential operator A0 of order m 0 such that A A 2 L1 is trace class.

Proof. Let A0 be as in Proposition 10.2.19. Then A A0 is compactly based and of order 1. A compactly based pseudo-differential operator of order 1 is a Shubin pseudo-differential operator of order 1,thatis,AA0 2 G1. By Theorem 10.2.20 the operator A A0 is trace class.

Theorem 10.2.20 is not quite the regularity that accords with Connes’ noncommutative spectral dimension. The next result is well known for compactly supported pseudo-differential operators, and the same statements for compactly based operators follow from the previous corollary.

Theorem 10.2.22. Let A be a compactly based pseudo-differential operator of order d d m.Ifm<0 then the extension A : L2.R / ! L2.R / is a compact linear operator. If m<d then the extension A is trace class.

The theorem follows from results in the literature, see the end notes. 328 Chapter 10 Preliminaries to the Applications

Example 10.2.23. If A 2 Lm1 is compactly based and B 2 Lm2 is compactly based, d and m1 C m2 Dd,thenAB, BA 2 L extend to compact linear operators with trace class difference, ŒA, B 2 L1.

The Noncommutative Residue

A smooth function p 2 C 1.Rd Rd / is a homogeneous function of order m in s except near zero if

˛mp.u, s/ D p.u, ˛s/, ˛>1, u 2 Rd , jsj1.

Deﬁnition 10.2.24. A pseudo-differential operator A 2 Lm, m 2 R, and its symbol p 2 S m are termed classical if p has an asymptotic expansion

X1 p pmj j D0

mj where each pmj 2 S is a homogeneous function of order mj in s except near zero. If needed, the set of classical pseudo-differential operators of order m is denoted by CLm.

Remark 10.2.25. The principal symbol of a classical pseudo-differenPtial operator A 1 of order m is the leading term pm in the asymptotic expansion p j D0 pmj .

Example 10.2.26. In Example 10.2.9 the operator .1 /m=2 2 Lm, m 2 R, had the symbol h2sim 2 Sm.Sincehsim Djsjm.1 Cjsj2 /m=2, jsj > 1,byusingthe binomial theorem the symbol of .1 /m=2 has the asymptotic expansion

X1 m j m2j h2si m j2sj . j D0 2

Hence, .1 /m=2 is classical with principal symbol .2/mjsjm for jsj1.

If A is a compactly based classical pseudo-differential operator of order m we canP take, without loss, each term in the asymptotic expansion of its symbol p 1 j D0 pmj to be a symbol with compact support in its ﬁrst variable. If m is an integer, the term pd makes sense (and is identically zero if m<d). If m is a non-integer, then we set pd 0. Let ds denote the volume element of the d 1-sphere

Sd1 :Dfs 2 Rd : jsjD1g. Section 10.2 Pseudo-differential Operators and the Noncommutative Residue 329

When d D 1thends is the counting measure on f1, 1g. As remarked earlier, we scale the usual deﬁnition of the noncommutative residue by the reciprocal of d.

Deﬁnition 10.2.27. The noncommutative residue of a classical compactly based pseudo-differential operator A is the scalar value Z Z 1 ResW .A/ :D pd .u, s/duds. d Sd 1 Rd

The noncommutativeresidue on operators includes integration of smooth compactly supported functions.

1 Rd d=2 Example 10.2.28. If 2 Cc . /,thenM.1 / is a compactly based classical pseudo-differential operator of order d. It has principal symbol .u/j2sjd , u 2 Rd , jsj1, which reduces to .2/d .u/ on the sphere jsjD1.Then Z Sd1 d=2 Vol . / ResW .M.1 / / D .u/du. (10.12) d.2/d Rd

The noncommutative residue restricted to operators of order d has properties typical of a singular trace.

Proposition 10.2.29. The noncommutative residue (a) is a linear functional on classical compactly based pseudo-differential operators of order d.

(b) vanishes on operators of order m<d (those that are trace class), that is, if A 2 CLm, m<d, is compactly based then

ResW .A/ D 0.

ResW .ŒA, B/ D 0.

Proof. (a) The linearity of ResW is evident from linearity of the principal symbol. (b) By construction pd 0ifm<d. (c) From Example 10.2.13 we have that ŒA, B 2 CLd1. The assertion now follows from (b).

In Chapter 11 we generalize the scalar noncommutative residue and show, as the above properties suggest, that the noncommutative residue is associated to a singular trace. This leads to a deﬁnition of the noncommutative residue for an arbitrary noncommutative geometry of Connes in Chapter 12. 330 Chapter 10 Preliminaries to the Applications

10.3 Pseudo-differential Operators on Manifolds

We indicate how pseudo-differential operators and the noncommutative residue are deﬁned on a closed Riemannian manifold.

A Brief Background on Closed Riemannian Manifolds Deﬁnition 10.3.1. A Hausdorff topological space X is said to be a smooth manifold (of dimension d) if there exists

(a) an open covering Ui , i 2 I , of X;

d (b) a collection of homeomorphisms hi : Ui ! R such that the mappings

1 hi ı hj : hj .Ui \ Uj / ! hi .Ui \ Uj / are smooth.

Each pair .Ui , hi / is called chart and the collection .Ui , hi /, i 2 I , is called an atlas. C 1 A function f : X ! is smooth if its image f ı hi is smooth within the chart Ui for each i 2 I . A smooth partition of unity f i gi2I is a set of smooth positive Pfunctions such that, for each fixed x 2 X, i .x/ 6D 0 for only finite number of i and i2I i .x/ D 1. The partition is subordinate to an atlas if supp. i / Ui , i 2 I . A (smooth n-dimensional) fiber bundle E ! X with fiber F is a smooth manifold 1 E with a surjection : E ! X such that: Ex :D .x/ is a manifold of dimension n (called the fiber over x 2 X); and there is a neighborhood U of x and a smooth diffeo- 1 morphism : .U / ! U F such that x : Ex !fxgF is a diffeomorphism (called a local trivialization). d d d The tangent space TuR Š R , u 2 R , is the vector space of directional deriva- d tives @.u/ for a tangent vector .u/ at u 2 R . The tangent bundle TX of a manifold X is the bundle with fibers .x, Tx X/ where TxX, x 2 Ui , i 2 I , is the tangent space Rd 1 Thi .x/ for x 2 Ui . The Jacobian Jij of the transition function hi ı hj , i, j 2 I , Rd Rd provides the transition functions between Thj .x/ and Thi .x/ when x 2 Ui \Uj . Hence, the charts of X provide the local trivializations for the tangent bundle. If the determinants det.Jij /, i, j 2 I , have the same sign, then X is called orientable. The cotangent bundle T X is the similarly constructed bundle with fibers .x, Tx X/where Tx X is the dual of TxX. d Rd Let fekgkD1 denote the standard basis of . For a chosen chart .U , h/ denote by @k.x/, k D 1, :::, d, the basis of TxX, x 2 U , corresponding to the standard partial Rd derivatives @ek , k D 1, :::, d,inTh.x/ .Letdxk.x/, k D 1, :::, d, denote the dual basis, i.e. dxk.x/[email protected]// D ıkm where ıkm is the Kronecker delta. A Riemannian metric is a collection of inner products

d gx : TxX TxX ! C Section 10.3 Pseudo-differential Operators on Manifolds 331 such that the coordinates of the metric in any chart .U , h/

gkm.x/ :D gx [email protected]/, @m.x// x 2 U , k, m D 1, :::, d C are smooth functions gkm : U ! . A Riemannian manifold .X, g/ is a manifold X with a Riemannian metric g.Let

d detg.x/ :D det.Œgkm.x/k,mD1/ denote the determinant of the positive definite d d matrix of the coordinates of a metric. The determinant detg is independent of the charts within a maximal atlas and is a strictly positive smooth function on X. The metric defines an isomorphism between TX and T X. The induced family of inner products on the cotangent bundle is denoted by g1. Given a metric g the co-sphere bundle SX over X is the bundle with fibers

1 Sx X Dfv 2 Tx X : gx .v, v/ D 1g. It is trivialized by a chart .U , h/ and S U Š U Sd1. A co-disc bundle D.r/X over X of radius r>0 is the bundle with ﬁbers

1 Dx .r/X Dfv 2 Tx X : gx .v, v/ rg. It is also trivialized by a chart .U , h/ and D.r/U Š U fs 2 Rd : jsjrg. Let f i gi2I be a smooth partition of unity subordinate to an atlas. The integral of a continuous compactly supported function f on an orientable Riemannian manifold .X, g/ is the linear functional Z X Z p 1 fdx :D . i f/ı hi .x/ detg.x/dx. X i2I hi .Ui / It is independent of the partition of unity and, up to constant, is the unique linear functional on the space of continuous compactly supported functions that agrees with the Lebesgue integral on each chart. By the Radon–MarkovR theorem, e.g. [188, The- orem IV.18], there exists a Radon measure associated to X dx, called the volume form, and it is in the sense of this measure that we define the equivalence classes Lp.X/,1 p 1. We note that L2.X/ is a separable Hilbert space. The condition of orientability is a convenience, neither the integral nor the Laplace– Beltrami operator defined in the next section require it. The integral on a non-oriented manifold can be defined using the Lebesgue density jdxj, which transforms as jdhi ı 1 1 hj xjDjdetJij .x/jjdxj instead of the usual change of coordinates dhi ı hj x D detJij .x/dx, [147, §14]. The integral defined using the Lebesgue density is still the linear functional which agrees with the Lebesgue integral on each chart. We will use orientability to simplify proofs, but the statements in the following sections are true for non-oriented as well as for oriented Riemannian manifolds. 332 Chapter 10 Preliminaries to the Applications

AmanifoldX is compact if every open cover of X admits a finite subcover. In this case an atlas with a finite number of charts can be chosen. Since det g is smooth and strictly positive, for a compact Riemannian manifold the spaces Lp.X/ (defined using g or any other Riemannian metric) are equivalent. A compact manifold is closed if hi .Vx/ is an open set for every neighborhood Vx Ui of x 2 Ui .Inthissensea closed manifold is compact and has no boundary.

Pseudo-differential Operators on Manifolds

For a chart .U , h/ of a closed Riemannian manifold X deﬁne an operator Wh : 1 d 1 1 C .R / ! C .U / by setting Whf :D f ı h. For a smooth function 2 C .X/ 1 1 denote by M : C .X/ ! C .X/ the bounded operator .M /.u/ :D .u/ .u/, 2 C 1.X/. The following deﬁnition is taken from [221].

Deﬁnition 10.3.2. An operator A : C 1.X/ ! C 1.X/ is called a pseudo-differential operator of order m if (a) for every chart .U , h/ and for all functions , 2 C 1.X/ supported in U , the 1 operator Wh MAM Wh is a pseudo-differential operator of order m; 1 (b) for all functions , 2 C .X/ with D 0, the operator MAM is an integral operator with smooth kernel. If needed we denote the linear space of pseudo-differential operators of order m on the manifold X by Lm.X/.

Example 10.3.3. Let X be a closed Riemannian manifold. There exists an operator 1 1 1 1 (see e.g. [38]) g : C .X/ ! C .X/ such that g : C .U / ! C .U / and

Xd q p 1 1 1 1 Rd .Wh g Wh/f D det.g /@m. detg.g /mk@kf/, f 2 C . / m,kD1 for every chart .U , h/. The operator g is a pseudo-differential operator of order 2 and is called the Laplace–Beltrami operator.

In a more general setting g denotes the Hodge–Laplacian operator restricted to smooth functions on X. However, we do not need to discuss the exterior bundle here. For scalar pseudo-differential operators, which we discuss in this book, the consideration of the Laplace–Beltrami operator sufﬁces. On a closed manifold the pseudo-differential operators obey identical properties to the compactly supported pseudo-differential operators on Rd .

Proposition 10.3.4. The pseudo-differential operators on a closed Riemannian manifold X of dimension d Section 10.3 Pseudo-differential Operators on Manifolds 333

(a) form a graded algebra with adjoint.

(b) have an extension (also denoted A) satisfying

A : H r .X/ ! H rm.X/, 8r 2 R

if A is order m (a simple deﬁnition of the Sobolev spaces on a closed manifold are given below).

m (c) have a closed extension if A is order m>0, A : H .X/ ! L2.X/.

(d) have a bounded extension if A is order m D 0, A : L2.X/ ! L2.X/. (e) have a compact extension if A is order m<0, which is trace class if A is order m<d.

A pseudo-differential operator A is classical if its restriction to a chart in Deﬁ- nition 10.3.2(a) is classical [221, §4]. The principal symbol of a classical pseudo- differential operator A of order m is deﬁned by the pullback on a local trivialization of the tangent bundle,

d pm.v/ :D qm.h.x/, s/, v 2 T U Š U R , where .U , h/ is a chart and qm is the principal symbol of the classical operator 1 Rd Wh M AM Wh on , .x/ D .x/ D 1, x 2 U . The principal symbol is independent of the chart and deﬁnes a locally homogeneous function on the tangent bundle T X [221, §4.3]. For a classical operator A of order m with pm.x, s/ > 0fors 6D 0, where, in an abuse of notation, we write v D .x, s/ 2 T X, and whose closed extension is m a positive operator A : H .X/ ! L2.X/, then one can deﬁne the powers of A, Ar , r 2 R, such that Ar is a classical pseudo-differential of order mr with principal r symbol .pm.x, s// [216], [221, §10].

r=2 Example 10.3.5. The pseudo-differential operator .1 g/ , r 2 R,isorderr, classical, and locally has principal symbol j2g1=2.x/sjr for jsj >, >0. For all r=2 r>0, .1g / has an extension to a positive operator on the Hilbert space L2.X/ r r=2 (with domain H .X/). For all r<0, .1g / has an extension to a compact linear operator on the Hilbert space L2.X/. The Sobolev spaces H r .X/, r 2 R, on a closed Riemannian manifold are conveniently viewed as

r r=2 H .X/ :Dff 2 L2.X/ : kf kH r :Dk.1 g / f k2 < 1g.

They are Hilbert spaces complete in the norm kkH r and the Sobolev norms for alternative Riemannian metrics are equivalent. 334 Chapter 10 Preliminaries to the Applications

The Noncommutative Residue on Manifolds We deﬁne the noncommutative residue of a classical pseudo-differential operator of order d where d is the dimension of the closed Riemannian manifold X.Letdv be the volume form of the manifold S X.

Deﬁnition 10.3.6. The noncommutative residue of a classical pseudo-differential operator A of order d is the scalar value Z 1 ResW .A/ :D pd .v/dv. d SX As in Example 10.2.28in the prior section, the noncommutativeresidue on operators includes integration of smooth functions.

1 d=2 Example 10.3.7. If 2 C .X/ then M.1g / is a classical pseudo-differential operator of order d. Locally, the principal symbol has the form .2/d .x/jg1=2.x/sjd , u 2 Rd , which reduces to .2/d .x/ on the sphere jg1=2.x/sjD1. Then, patching together the local contributions, Z Vol .Sd1/ Res .M .1 /d=2/ D .x/dx. (10.13) W g d d.2/ X In the form given in Deﬁnition 10.3.6 the noncommutative residue is overtly independent of a chosen atlas. We will show in the next chapter, and it is well known, that the noncommutative residue is equivalently deﬁned using the simpler formula on Sd1 X , Z 1 ResW .A/ :D pd .x, s/dxds, d XSd 1 which implicitly chooses an atlas.

10.4 Notes

Symbols and Pseudo-differential Operators There are many classic references on pseudo-differential operators, including the essential texts [115,221], which contain detailed treatment and discussion of what has been summarized in this section. Chapter 2 of [201] contains an uncomplicated introduction to uniform symbols. There are many variants of the symbol classes, including general Hörmander symbol classes [115], and variants of the Shubin classes on Rd [221], which are unnecessary for the following chapters. The pseudo-differential operators we introduce correspond to the uniform class of type (1,0) [201, Chapter 2], [221, Problem 3.1]. For the proof of Proposition 10.2.3 see [201, Theo- rem 2.1.6, §2.2, Remark 2.5.5]. For Proposition 10.2.5 see [201, Proposition 2.2.4]. For Theo- rem 10.2.7 on the unique symbol for uniform differential operators see [201, Theorem 2.5.6]. Section 10.4 Notes 335

For Proposition 10.2.11 on the asymptotic formula linking the symbol and amplitude see [201, Theorem 2.5.8].

Calculus and Regularity of Pseudo-differential Operators Proposition 10.2.15 is the ubiquitous statement of the pseudo-differential calculus. For the proof for uniform symbols see [201, Section 2.5]. A more general consideration is the notion of a proper pseudo-differential operator [221, §3]. The zero order proper pseudo-differential d d operators do not deﬁne bounded operators from L2.R / ! L2.R / generally, nor compact operators, see [221, §6, §7]. The boundedness properties of uniform symbols make them a simpler object for the purposes of functional analysis. Boundedness properties of zero order operators has a long history, see Calderón and Vaillancourt [25], Cordes [56], and Kato [132]. For the regularity theorem, Theorem 10.2.14, see [201, Theorem 2.6.11]. For a more detailed investigation see [221, §7].

Compactly Supported Pseudo-differential Operators A treatmentof compactlysupported pseudo-differentialoperators can be found in Shubin [221, §6]. However, the exposition in most texts favor the class Gm as a study of pseudo-differential operators on Rd that have Hilbert–Schmidt or trace class extensions [221, IV]. The association of the class Gm to quantization, Weyl symbols, and anti-Wick symbols, makes the class Gm natural to study. Theorem 10.2.20 is contained in [221, Section 27]. However, in terms of Connes’ spectral dimension [49, Section 2], [33], the compactly based pseudo-differential operators have the more ﬁtting trace class properties. Theorem 10.2.22 can be obtained directly from general results in [5], or by using the Weyl symbol as in [198], or by using the anti-Wick symbol as in [15,24].

The Noncommutative Residue The notion of a classical pseudo-differential operator is universal in every text on pseudo- differential operators [221, §3.7], [201, §2.5.3]. Classical symbols are a direct generalization of polynomials in s, making a classical pseudo-differential operator the direct generalization of a differential operator. The noncommutative residue originated in [257, Section 7], although the form given of the integral over the trivial sphere bundle was co-incidental to the original study of the residue of -functions associated to complex powers of elliptic pseudo-differential operators. More extensive results on manifolds and that the residue arises from a density are shown in [258]. The properties of the noncommutative residue in Proposition 10.2.29 were noticed by Wodzicki in [257, Final Remarks 7.1.3]. The noncommutative residue deﬁned by Wodzicki generalized the study by Adler [2], and Manin [160].

Pseudo-differential Operators on Manifolds An introduction to smooth Riemannian manifolds can be found in [147]. The reference used for pseudo-differential operators on manifolds is Shubin’s monograph [221]. The proof of Proposition 10.3.4 follows from [221, Theorem 7.3] and [221, §6.4]. As mentioned above, Wodzicki deﬁned the noncommutative residue in [257, Final Remarks 7.1.3]. See also the treatment of the noncommutative residue in [100]. Chapter 11 Trace Theorems

11.1 Introduction

This chapter uses Nigel Kalton’s notion of a modulated operator, and the Banach algebra of modulated functions, to extend the noncommutative residue of Mariusz Wodz- icki on classical pseudo-differential operators of order d to a wider class of integral operators, the class of Laplacian modulated operators, and to prove that the value of a Dixmier trace applied to a Laplacian modulated operator is calculated by the noncommutative residue. More precisely, the extension of the scalar valued noncommutative residue of Wodzicki to compactly supported Laplacian modulated operators is vector- valued, and what we prove is that the value of a Dixmier trace applied to a compactly supported Laplacian modulated operator is given by an extended limit applied to the vector-valued noncommutative residue. d The exact deﬁnition of a Laplacian modulated operator T 2 L.L2.R // is that

1=2 d=2 1 kT kmod :D sup t kT.1 C t.1 / / k2 < 1, (11.1) t>0 P d @2 Rd where D iD1 2 , s D .s1, :::, sd / 2 , is the Laplacian operator and kk2 is @si d d=2 the Hilbert–Schmidt norm on L2.L2.R //. The operator .1 / should be read d as its extension to a positive bounded operator on L2.R /, as in Example 10.2.16. In a concrete sense the deﬁnition in (11.1) is not enlightening. A Laplacian mod- d ulated operator T is a Hilbert–Schmidt operator on L2.R /, therefore, as an integral d d operator, it has a symbol pT 2 L2.R R / such that Z 2ihu,si d .T x/.u/ D e pT .u, s/.Fx/.s/ds, x 2 L2.R /. Rd

Here F denotes the Fourier transform. We observe in this chapter that the set of Lapla- cian modulated operators forms a Banach space, which is isometric to the Banach space of square integrable symbols such that Z Z 1=2 d=2 2 kpT kmod :D sup t jpT .u, s/j duds < 1. t>0 jsj>t Rd

This is a convenient condition by which to understand which integral operators are Laplacian modulated, and makes it evident that all compactly based pseudo- Section 11.1 Introduction 337 differential operators of order d are Laplacian modulated. Recall that an operator d T 2 L.L2.R // is compactly based if MT D T for some compactly supported 1 Rd smooth function 2 Cc . /, and compactly supported if M TM D T for some 1 Rd 1 Rd compactly supported smooth function 2 Cc . /.HereM , 2 Cc . /,denotes either the zero order operator given by the pointwise product, .M x/.u/ :D 1 Rd .u/x.u/, x 2 Cc . /, see Section 10.2, or its extension to a bounded operator, d .Mx/.u/ :D .u/x.u/, x 2 L2.R /. If a bounded operator T is Laplacian modulated, and compactly based, with symbol pT , we show that its vector-valued noncommutative residue is well deﬁned by the formula Z Z 1 1 Res.T / :D pT .u, s/duds , (11.2) log.2 C n/ jsjn1=d Rd nD0 where Œ denotes the equivalence class in l1=c0. Note that an extended limit ! on l1 (a state on l1 that vanishes on c0) is a state on the algebra l1=c0. The main result of this chapter is the following trace theorem.

Theorem 11.1.1 (Trace Theorem). Every compactly supported Laplacian modulated operator T belongs to L1,1 and

(a) Tr!.T / D !.Res.T // (11.3)

for any Dixmier trace Tr!.

(b) Tr!.T / D Res.T / (11.4)

for all Dixmier traces Tr! if and only if Res.T / is scalar valued.

pT .u, s/duds D Res.T / log.n/ C O.1/, n 1. jsjn1=d Rd

Here, diag is the diagonal operator for any chosen basis of the Hilbert space d L2.R /. Recall that a trace ' on L1,1 is referred to as normalized if '.diagf.1 C 1 1 n/ gnD0/ D 1. We derive Connes’ original trace theorem from Theorem 11.1.1 in Section 11.5. 1 Rd 1 Rd We state Connes’ theorem as follows. If A : Cc . / ! Cc . / is a compactly supported classical pseudo-differential operator of order d,andResW denotes the noncommutative residue of Wodzicki, then (the extension) A 2 L1,1 and

'.A/ D ResW .A/ 338 Chapter 11 Trace Theorems

for every normalized trace ' on L1,1. Section 11.5 also gives an example of a com- 1 Rd 1 Rd pactly supported pseudo-differential operator A : Cc . / ! Cc . / of order d such that Res.A/ is not scalar valued, which implies that the value Tr!.A/ depends on the extended limit !. Hence, there are non-classical pseudo-differential operators that are not measurable operators in Connes’ sense. Finally, Theorem 11.1.1 allows us to integrate square integrable functions using a singular trace and an operator calculus. This does not follow trivially from the ability to integrate smooth functions as seen in d Example 10.2.28 and Example 10.3.7. If f 2 L2.R / is compactly supported and Mf 1 Rd is the normal operator densely defined by .Mf x/.u/ :D f .u/x.u/, x 2 Cc . /, d=2 then Mf .1 / 2 L1,1 and Z Sd1 d=2 Vol. / '.Mf .1 / / D f.u/du d.2/d Rd for every normalized trace ' on L1,1, see Theorem 11.7.5. A version of Theorem 11.1.1 for closed manifolds is provided in Section 11.6. A Hodge–Laplacian operator on a d-dimensional closed Riemannan manifold X with metric g is defined by replacing the Laplacian with the Laplace–Beltrami operator g in equation (11.1). Generally, a Hodge–Laplacian modulated operator does not have a global symbol. Instead, for a Hodge–Laplacian modulated operator T : L2.X/ ! L2.X/,wedefine 1 Xn 1 Res.T / :D hTe , e i , log.2 C n/ j j j D0 nD0

1 where Œ denotes the equivalence class in l1=c0,andfej gj D0 is an orthonormal ba- 2 sis of eigenvectors of the Laplace–Beltrami operator such that g ej D j ej and 2 2 0 1 are increasing. That such a basis exists is a well known result of Hodge Theory and spectral geometry. With this deﬁnition of the noncommutative residue the manifold version of Theorem 11.1.1 follows. The critical feature of this, completely spectral, residue is that it is an extension of the noncommutative residue of Wodzicki.

Corollary 11.1.2 (Spectral formula of the scalar noncommutative residue). If A : C 1.X/ ! C 1.X/ is a classical pseudo-differential operator of order d on a closed d-dimensional Riemannian manifold X, and ResW is the noncommutative residue of Wodzicki, then

1 Xn ResW .A/ D lim hAej , ej i, n!1 log.2 C n/ j D0

1 where fej gj D0 is an ordered eigenbasis, as above, of the Laplace–Beltrami operator. Section 11.2 Modulated Operators 339

The corollary provides a fully spectral view of the noncommutative residue in the differential geometry of closed manifolds. The door is therefore open to extend the concept of the noncommutative residue, and the trace theorems, to noncommutative versions of spectral geometry such as Connes’ noncommutative geometry; which we do in Chapter 12.

11.2 Modulated Operators

The context, and the mechanism, by which we prove Theorem 11.1.1 is Nigel Kalton’s notion of a modulated operator. We deﬁne modulated operators in this section and prove their properties, which underlie the subsequent sections. Modulated operators originated with the following problem. We showed in Sec- tion 7.5 that, in general, expectation value formulas fail for Dixmier traces. Kalton introduced left ideals of Hilbert–Schmidt operators, called modulated operators, where partial expectation value formulations apply. Theorem 11.2.3 below represents the desired outcome that motivated their introduction. In the following, and throughout the subsequent sections, H is a separable Hilbert space and diag is the diagonal operator for any arbitrary basis of H .

Deﬁnition 11.2.1. Let V 2 L.H / be a positive operator. An operator T 2 L.H / is called V -modulated if

1=2 1 kT kmod :D sup t kT.1 C tV / k2 < 1, t>0 where kk2 denotes the Hilbert–Schmidt norm on L2. Note that every V -modulated operator is automatically Hilbert–Schmidt since

kT k2 .1 CkV k1/kT kmod. Proposition 11.2.2. Let V 2 L.H / be a positive operator. The set of all V -modulated operators is a Banach space. For every A 2 L.H / and for every V -modulated operator T , we have kAT kmod kAk1 kT kmod. Proof. The only statement needing to be proved is the completeness of the linear space of V -modulated operators. Let Tn, n 2 N, be a Cauchy sequence in the space of all V -modulated operators. Note that kTnkmod is a Cauchy sequence in R and that 1 1 Tn.1 C V/ is a Cauchy sequence in L2.Hence,Tn.1 C V/ ! S 2 L2. Setting 1 1 T :D S.1 C V/,wehaveTn.1 C tV / ! T.1 C tV / in L2. Hence, for every t>0, we have

1=2 1 1=2 1 t kT.1 C tV / k2 D lim t kTn.1 C tV / k2 lim kTnkmod. n!1 n!1 Taking the supremum over t>0, we infer that T is V -modulated. 340 Chapter 11 Trace Theorems

Our main result on modulated operators is the following theorem on the expectation values of a V -modulated operator with respect to an eigenbasis of V 2 L1,1.The proof of the theorem is given further below. By the term strictly positive we mean positive with trivial kernel. The commutator subspace of L1,1, denoted Com.L1,1/, was introduced in Section 5.7.

Theorem 11.2.3. Let 0

Corollary 11.2.4. Let 0

(a) for every trace ' : L 1 ! C, 1, 1 '.T / D ' ı diag fhTen, enignD0 .

(b) for every Dixmier trace Tr! : M1,1 ! C, 1 Xn Tr .T / D ! hTe , e i . ! log.2 C n/ j j j D0

(d) the formula Xn hTej , ej iDc log.n/ C O.1/, n 1, j D0

holds if and only if '.T / D c for every normalized trace ' : L1,1 ! C. Section 11.2 Modulated Operators 341

1 L Proof. (a) From Theorem 11.2.3 we have that T diagfhTen, enignD0 2 Com. 1,1/. Since all traces vanish on the commutator subspace the result follows. (b), (c), (d) From Theorem 11.2.3 we have 1 Xn 1 Xn 1 .j , T/D hTe , e iCO . log.2 C n/ log.2 C n/ j j log.2 C n/ j D0 j D1

Since ! vanishes on c0 then (c) and (d) follow from Theorem 10.1.3, and (b) follows from Theorem 10.1.2 (a).

We prove Theorem 11.2.3.

Lemma 11.2.5. Let V 2 L.H / be a positive operator. An operator T 2 L2 is V - modulated if and only if

1 1=2 kTEV Œ0, t k2 D O.t /, t>0, where EV is the spectral measure of V .

1 1 Proof. Observe that EV Œ0, t 2.1 C tV / for t>0 and, therefore,

1 1 1=2 kTEV Œ0, t k2 2kT.1 C tV / k2 2kT kmod t .

In order to prove the converse assertion, assume for simplicity that V 1. Let t 2 Œ2k,2kC1/ for some k 0.

Xk1 1 k j 1 j 1 kT.1 C tV / k2 kTEV Œ0, 2 k2 C kTEV .2 ,2 .1 C tV / k2 j D0 Xk1 1=2 j 1 1 j 1 j O.t / C .1 C t2 / kTEV .2 ,2 k2 j D0 Xk1 O.t1=2/ C O.1/ .1 C 2kj 1/1 2j=2 D O.t1=2/. j D0

Lemma 11.2.6. Let V 2 L.H / be a positive operator and let A 2 L.H /. If T is V -modulated, then TA is jVAj-modulated.

n Proof. Without loss of generality, kV k1 1andkAk1 1. Let pn D EV Œ0, 2 n and qn D EjVAjŒ0, 2 for n 0. We have

j j k j=2 k.1 pj /Aqkk1 2 kVAqkk1 D 2 , kTpj k2 const 2 . 342 Chapter 11 Trace Theorems

Therefore,

Xk kTAqkk2 kTpkAqkk2 C kT.pj 1 pj /Aqkk2 j D1 Xk kTpkk2 C kTpj 1k2 k.1 pj /Aqkk1 j D1 Xk const 2k=2 C 2j=2k D O.2k=2/. j D1

The assertion follows from Lemma 11.2.5.

Remark 11.2.7. Let 0 V1 V2 2 L.H /. If T 2 L.H / is V1-modulated, then T is also V2-modulated.

Lemma 11.2.8. Let V 2 L1,1 be a positive operator. Then V is V -modulated.

Proof. Since V 2 L2,itfollows,fort>0, that X1 1 2 2 1 2 n1 1 n 1 kVEV Œ0, t k2 D Tr.V EV Œ0, t / D Tr.V EV .2 t ,2 t // nD0 X1 2n 2 n1 1 2 t Tr.EV Œ2 t , 1// nD0 X1 2n 2 n1 1 D 2 t nV .2 t /. nD0

1 Recall that V 2 L1,1 and, therefore, nV .s/ const s , s>0. Hence, X1 1 2 2n 2 nC1 1 kVEV Œ0, t k2 const 2 t 2 t D O.t /. nD0

Thus, V is V -modulated by Lemma 11.2.5.

Lemma 11.2.9. Let 0 2, we have TV 2 Lp=.p1/,1. Section 11.2 Modulated Operators 343

Proof. Let S :D TV1=p and let 2k n<2kC1. By Lemma 11.2.5, we have X1 X1 1 j 1 j .j C1/=p j 1 j SEV 0, kSEV .2 ,2 k2 2 kTEV .2 ,2 k2 n 2 j Dk j Dk X1 const 2.j C1/=p 2j=2 D const 2k.1=p1=2/ D O.n1=p1=2/. j Dk

1 If fengnD0 is an eigenbasis for V ordered so that Ven D .n, V/en, n 0, set

Xn1 Snx :D hx, ej iSej , x 2 H . j D0

We have X1 2 2 2 .j , S/ D minfkS Ak2 :rank.A/ ngkS Snk2 j Dn X1 2 2 1 2=p1 D kSej k SEV 0, D O.n /. n j Dn 2

Hence, .n, S/ D O.n1=p1/ and the assertion follows.

L 1=p L Lemma 11.2.10. Let 0

Xn Xn .k,

Here

Proof. Let fn, n 0, be a basis such that .

jTr..

krn qnk1 k.

1=p On the other hand, we have TV 2 Lp=.p1/,1. Therefore,

1=p 1=p jTr.T .rn pn//jDjTr.T .rn pn//jDjTr..rn pn/.T V / V .rn pn//j 1=p 1=p k.rn pn/.T V /k1 kV .rn pn/k1 1=p 1=p krn.T V /k1 kV .1 pn/k1 2XnC1 .k, TV1=p/ .n, V/1=p D O.1/. kD0 Therefore, 1 1 jTr..

1=p Proof of Theorem 11.2.3. It follows from Lemma 11.2.9 that TV 2 Lp=.p1/,1. In particular, T 2 L1,1. Applying Lemma 11.2.9 to the V -modulated operator 1 diag.fhTen, enignD0/, we obtain (a). Applying Lemma 11.2.10 to the operators T and iT ,forn 0 we obtain

Xn Xn Xn Xn .k,

Xn Xn .k, T/ hTek , ekiDO.1/. (11.6) kD0 kD0 1 Applying (11.6) to the V -modulated operator T diag.fhTen, enignD0/, we obtain (b). Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue 345

11.3 Laplacian Modulated Operators and Extension of the Noncommutative Residue

In this section we consider operators that are modulated with respect to the bounded positive operator .1/d=2,where is the Laplacian on Rd . We call such operators Laplacian modulated operators. We show that the Laplacian modulated operators are integral operators on Rd that extend pseudo-differential operators of order d and that there is a vector-valued residue for Laplacian modulated operators which extends Wodzicki’s noncommutative residue.

The Banach Algebra of Modulated Functions We begin by introducing the algebra of modulated functions. They are an interesting class of functions and we discuss their features; they will be used in subsequent sections.

d Deﬁnition 11.3.1. The algebra Lmod;1.R / of modulated functions consists of those d functions f 2 L1.R / for which Z d kf kLmod;1 :D sup.1 C t/ jf.s/jds < 1. t>0 jsj>t

Convolution is the multiplication in this algebra. This is shown in the following lemma.

d d Lemma 11.3.2. If f , g 2 Lmod;1.R /, then f?g2 Lmod;1.R /. Proof. We have, for t>0, Z Z Z Z jg.u/jjf.s u/jduds jg.u/jjf.s u/jduds jsjt jujjsj=2 Z Rd jujZt=2 D jf.s/jds jg.u/jdu D O.td /. Rd jujt=2

Now if juj < jsj=2andjsjt,thenjs ujjsj=2 t=2. Hence, Z Z “ jg.u/jjf.s u/jduds jg.u/jjf.s u/jduds jsjt“juj

These estimates show that Z sup.1 C t/d jf?g.s/jds t>0 jsj>t Z Z sup.1 C t/d jg.u/jjf.s u/jduds < 1 t>0 jsj>t Rd by splitting the second integral into the regions juj < jsj=2andjujjsj=2.

d Lemma 11.3.3. Lmod;1.R / is a Banach space.

d Proof. Observe, from Deﬁnition 11.3.1, that Lmod;1.R / is a normed space. If fn, d d n 0, is a Cauchy sequence in Lmod;1.R /, then it is a Cauchy sequence in L1.R /. d Thus, it converges to f in L1.R /.Foragiven>0, select N sufﬁciently large so that Z d .1 C t/ jfn.s/ fm.s/jds jsjt for every m, n>Nand for every t>0. Thus, for every t>0 and for every n>N, we have Z Z d d .1 C t/ jfn.s/ f.s/jds D lim .1 C t/ jfn.s/ fm.s/jds . !1 jsjt m jsjt

Since >0 is arbitrarily small, the assertion follows.

d Theorem 11.3.4. Lmod;1.R / is a Banach algebra.

d Proof. The space of functions Lmod;1.R / is a Banach space and an algebra. In fact, it is proved in Lemma 11.3.2 that

kf?gkLmod;1 const.kf k1kgkLmod;1 Ckgk1kf kLmod;1/ constkgkLmod;1 kf kLmod;1 .

d Hence, Lmod;1.R / is a Banach algebra.

Theorem 11.3.4 shows that the modulated functions form a Banach algebra that is d a subalgebra of the integrable functions L1.R /. The following result shows that the algebra of modulated functions and the algebra of integrable functions are generated quite differently.

Lemma 11.3.5. The set of compactly supported integrable functions is not dense in d Lmod;1.R /. Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue 347

2d d d Proof. Let f.s/ :D .1 Cjsj/ , s 2 R ,andletg 2 L1.R / be a compactly supported function. Fix t sufﬁciently large so that g.s/ D 0forjsj >t. Thus, Z Z .1 C t/d jf gj.s/ds D .1 C t/d .1 Cjsj/2d ds jsj>t Zjsj>t 1 D const t d .1 Cjsj/2d jsjd1djsjconst. t Thus, the distance from f to the set of all compactly supported functions is positive.

d Deﬁnition 11.3.6. Deﬁne the Banach space Lmod;2.R / consisting of those functions f 2 L .Rd / with jf j2 2 L .Rd / and norm kf k :Dkjf j2k1=2 . 2 mod;1 Lmod;2 Lmod;1

d Since each function f 2 Lmod;2.R / is square integrable it is interesting to know d whether the Fourier transform .Ff/belongs to Lmod;2.R /. The answer is negative.

Example 11.3.7. For every ˛ 2 .1=4, 0/, we have

˛ 2 ˛d=2 d f˛.u/ :Djuj K˛.2juj/, .Ff˛/.u/ D const .1 Cjuj / , u 2 R , where K˛ is the Macdonald function (a modiﬁed Bessel function of the second kind d d [1]) for ˛. In particular, f˛ 2 Lmod;2.R / and .Ff˛/ … Lmod;2.R /.

Proof. We conﬁrm that the functions are related by the Fourier transform. Applying a unitary transform, we obtain, for u 2 Rd , Z Z e2ihs,ui e2is1juj ds D ds. Rd .1 Cjsj2/˛Cd=2 Rd .1 Cjsj2/˛Cd=2

2 1=2 After substitution sk D vk.1 C s1 / ,2 k d, we obtain Z Z 2ihs,ui 2is1juj e e 2 ˛d=2 ds D .1 Cjvj / ds1dv Rd .1 Cjsj2 /˛Cd=2 Rd .1 C s2/˛C1=2 Z 1 Z 1 2is1juj e 2 ˛d=2 D 2 ds1 .1 Cjvj / dv ˛C1=2 Rd 1 1 .1 C s1 / ˛ D const juj K˛.2juj/.

d d Since Ff˛ 2 L2.R /, it follows that f˛ 2 L2.R /. Now, observe that Z Z 2 4˛2d 4˛d j.Ff˛/.s/j ds > const jsj ds D const .1 C t/ . jsj>t jsj>t 348 Chapter 11 Trace Theorems

2 d Hence .Ff˛/ 62 Lmod;1.R /. However, as the Macdonald function K˛ decreases 2 Rd exponentially at C1 [1], it follows that f˛ 2 Lmod;1. /. The situation is not improved even if we take a modulated function of compact support.

Example 11.3.8. We have

F d=2 Rd f.u/ :D juj1, . f /.u/ Djuj Jd=2.2juj/, u 2 ,

Rd where Jd=2 is a Bessel function of the ﬁrst kind [1]. In particular, f 2 Lmod;2. / is d a compactly supported function and .Ff˛ / … Lmod;2.R /. Proof. We conﬁrm that the functions are related by the Fourier transform. For s 2 Rd Rd1 2 1=2 with jsj1, we set v D .s2, :::, sd / 2 . Observe that jvj.1 s1 / and, for u 2 Rd , Z Z Z f.s/e2ihs,uids D e2ihs,uids D e2is1jujds Rd j j j j Z s 1 Z s 1 1 2is1juj D e dv ds1 1 jvj.1s2/1=2 Z 1 1 2 .d1/=2 2is1juj D const .1 s1 / e ds1. 1

d=2 The latter integral is juj Jd=2.2juj/. Evidently, the compactly supported function f 2 is modulated. Since d C 1 J .2juj/ juj1=2 cos 2juj , juj!1, d=2 4 then .Ff/is strictly of order juj.1Cd/=2. It follows that .Ff/2 is not modulated.

Laplacian Modulated Operators

d We associate the Banach algebra Lmod;1.R / to the notion of a Laplacian modulated operator.

d d Lemma 11.3.9. Every Hilbert–Schmidt operator T : L2.R / ! L2.R / can be represented in the form Z 2ihu,si d .T x/.u/ D e pT .u, s/.Fx/.s/ds, u 2 R , (11.7) Rd

d d for a unique function pT 2 L2.R R /. Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue 349

Proof. The operator T F1 is Hilbert-Schmidt, and is hence an integral operator with a square integrable kernel, [222, p. 23]. The assertion follows by applying the Plancherel theorem.

Deﬁnition 11.3.10. We call the function pT in equation (11.7) the symbol of the operator T . The relationship between the kernel and symbol of a Hilbert–Schmidt operator is Z 2ihuv,si d KT .u, v/ D e pT .u, s/ds, u, v 2 R . (11.8) Rd The symbol can be recovered from the operator as follows.

d d Lemma 11.3.11. Let T : L2.R / ! L2.R / be a Hilbert–Schmidt operator. If , >0, is a molliﬁer, as introduced in Section 10.2, then almost everywhere 2ihu,si 1 2ih,si d pT .u, s/ D lim e T..F /e /.u/, u, s 2 R #0

Proof. We have, for u 2 Rd ,that Z 2ihs,i 2ihu,vi 2ihs,ui T..F/e /.u/ D e pT .u, v/.v s/dv ! e pT .u, s/ Rd as # 0. Let be the Laplacian on Rd . The bounded positive operator .1 /d=2 : d d L2.R / ! L2.R / from Example 10.2.16 is used in the following deﬁnition.

d d Definition 11.3.12. A bounded operator T : L2.R / ! L2.R / is said to be Lapla- cian modulated if it is .1 /d=2-modulated, that is, 1=2 d=2 1 sup t kT.1 C t.1 / / k2 < 1, (11.9) t>0 d where kk2 denotes the Hilbert–Schmidt norm on L2.L2.R //. Observe that every Laplacian modulated operator is necessarily Hilbert–Schmidt, and as such has a unique symbol. The above definition is not enlightening in practice, however, Laplacian modulated operators are equivalently defined by the following property of their symbols.

d d Lemma 11.3.13. A Hilbert–Schmidt operator T : L2.R / ! L2.R / with symbol p is Laplacian modulated if and only if T Z 2 d jpT .u, /j du 2 Lmod;1.R / Rd d where Lmod;1.R / is the Banach algebra of modulated functions. 350 Chapter 11 Trace Theorems

Proof. It follows from Lemma 11.2.5 that T is Laplacian modulated if and only if

1 1=2 kTE.1/d=2Œ0, t k2 D O.t /, t>0. (11.10)

The spectral projection of the operator .1 /d=2 is given by the formula Z 1 F 2ihu,si Rd .E.1/d=2 Œ0, t x/.s/ D . x/.u/e du, s 2 , .1C4 2juj2/d=2t 1

Deﬁne a family of projections pt , t>0, by setting Z 2ihu,si d .pt x/.s/ :D .Fx/.u/e du, s 2 R . juj>t

There exist constants c1 and c2 such that, for every t 1, we have

1 1=d d=2 1=d p.c1t/ E.1/ Œ0, t p.c2t/ .

It follows from (11.10) that T is Laplacian modulated if and only if kTpt k2 D O.td=2/. The assertion now follows from the equality Z Z 2 2 kTpt k2 D jpT .u, s/j duds, t>0. jsj>t Rd Remark 11.3.14. The set of symbols of Laplacian modulated operators form a Ba- nach space S mod with norm Z Z 1=2 d=2 2 kpT kmod :D sup.1 C t/ jpT .u, s/j duds . t>0 jsj>t Rd In fact, Lemma 11.3.13 and its proof show that there is an isometry between the Ba- nach space of Laplacian modulated operators (see Proposition 11.2.2)and the Banach space of their symbols Smod.

d d Deﬁnition 11.3.15. An operator T : L2.R / ! L2.R / is said to be 1 Rd (a) compactly supported if there exist , 2 Cc . / such that MTM D T . 1 Rd (b) compactly based if there exists 2 Cc . / such that M T D T . The proof of the following lemma, as with its equivalent in Section 10.2, is omitted.

d d Lemma 11.3.16. A Laplacian modulated operator T : L2.R / ! L2.R / is

(a) compactly supported if and only if its kernel KT is compactly supported.

(b) compactly based if and only if its symbol pT is compactly supported in the ﬁrst variable. Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue 351

The Banach space of Laplacian modulated operators is an extension of the set of compactly based pseudo-differential operator of order d.

1 Rd 1 Rd Theorem 11.3.17. If A : Cc . / ! Cc . / is a compactly based pseudo-differential operator of order d, then (the extensions of) both A and A are Laplacian modulated.

Proof. The symbol p of a pseudo-differential operator A of order d belongs to S d . In particular, jp.u, s/jconst hsid .

The symbol of a compactly based operator is compactly supported in the ﬁrst variable. Fix a compact set K Rd such that p.u, s/ D 0foru … K.Wehave Z Z Z jp.u, s/j2duds const m.K/ hsi2d ds D O.td /, jsjt Rd jsjt where m is Lebesgue measure. It follows from Proposition 11.3.13 that A is Laplacian modulated. It follows from Proposition 10.2.19 that A D A1 C A2 with A1 being a compactly supported operator of order d and A2 being a compactly based operator of order 1. Hence, by the above, A1 , a compactly supported operator of order d,isaLaplacian 1 modulated operator. The operator A2 belongs to the Shubin class G .ThenA2 2 1 G .Ifq is a symbol of A2 ,then

q.u, s/ const .1 CjujCjsj/2d , u, s 2 Rd .

Observe that Z Z 1 .1 CjujCjsj/4d du D const .1 CjujCjsj/4d jujd1djuj Rd 0 Z 1Cjsj const .1 Cjsj/4d jujd1djuj 0 Z 1 C juj3d1djuj 1Cjsj D const .1 Cjsj/3d .

3d d d The mapping s ! .1 Cjsj/ , s 2 R , belongs to Lmod;1.R /. It follows from Lemma 11.3.13 that A2 is Laplacian modulated. Hence, A D A1 C A2 is Laplacian modulated. 352 Chapter 11 Trace Theorems

Residues of Laplacian Modulated Operators We deﬁne in this section the vector-valued residue of a compactly based Laplacian modulated operator. We show it is an extension of the noncommutative residue of a classical compactly based pseudo-differential operator of order d.

d d Proposition 11.3.18. If T : L2.R / ! L2.R / is a compactly based Laplacian modulated operator with symbol pT ,then Z Z 1 pT .u, s/duds D O.1/, n 0. log.2 C n/ jsjn1=d Rd

Proof. It follows from Lemma 11.3.16 that the symbol pT of T is compactly sup- d ported in its ﬁrst variable. Fix a compact set K R such that pT .u, s/ D 0for u … K. Observe that, for k 0,

m.K fs 2 Rd : ek jsjekC1g/ D const m.K/ekd.

Using the Cauchy–Schwartz inequality, we have Z Z Z Z

jpT .u, s/jduds D jpT .u, s/jduds k kC d k kC e jsje 1 R e jsje 1 K Z const m.K/ekd ekjsjekC1 Z 1=2 2 jpT .u, s/j duds Rd 1=2 const m .K/kpT kmod.

Similarly, Z Z 1=2 jpT .u, s/jduds const m .K/kpT kmod. jsj1 Rd It follows that ˇ Z Z ˇ Z Z ˇ ˇ ˇ ˇ ˇ pT .u, s/dudsˇ jpT .u, s/jduds jsjt Rd jsj1 Rd ŒlogX.t/ Z Z C jpT .u, s/jduds ek jsjekC1 Rd kD0 1=2 .2 C Œlog.t// const m .K/kpT kmod.

The assertion follows by substituting into this inequality the value t D n1=d , n 0, and the fact that log.n1=d / D d 1 log.n/. Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue 353

Deﬁnition 11.3.19. Let T be a Laplacian modulated operator and let pT denote its symbol. The vector-valued linear function Res from the set of all compactly based Laplacian modulated operators into l1=c0 deﬁned by setting Z Z 1 1 Res.T / :D pT .u, s/duds , (11.11) log.2 C n/ jsjn1=d Rd nD0 where Œ denotes an equivalence class in l1=c0, is called the residue. We show that the map Res extends the noncommutative residue.

Lemma 11.3.20. The residue of a compactly based pseudo-differential operator of order d depends only on its principal symbol.

Proof. If A1 and A2 are compactly based pseudo-differential operators of order d with the same principal symbol, then A1 A2 is a compactly based pseudo-differential operator with symbol p 2 S d1. In particular,

jp.u, s/jconst hsid1.

The symbol of a compactly based operator is compactly supported in the ﬁrst variable. Fix a compact set K Rd such that p.u, s/ D 0foru … K.Wehave ˇ Z Z ˇ Z ˇ ˇ ˇ ˇ 1 ˇ p.u, s/dudsˇ const m.K/ hsi d ds < 1. Rd Rd Rd

Hence, Res.A1 A2/ D 0 and, therefore, Res.A1/ D Res.A2/.

We understand the ﬁeld of scalars to be embedded in `1=c0 as the equivalence C 1 classes of convergent sequences, that is, if ˛ 2 then ˛ ŒfangnD0 where limn!1 an D ˛.

1 Rd 1 Rd Proposition 11.3.21. If A : Cc . / ! Cc . / is a classical compactly based pseudo-differential operator of order d, then its extension A is Laplacian modulated and

Res.A/ D ResW .A/.

Here ResW denotes the noncommutative residue of Wodzicki as deﬁned in Deﬁni- tion 10.2.27.

Proof. Let p be the principal symbol of the operator A. By Lemma 11.3.20 the value Res.A/ is determined by the equivalence class of the sequence Z Z 1 p.u, s/duds, n 0. log.2 C n/ jsjn1=d Rd 354 Chapter 11 Trace Theorems

Since p is homogeneous of order d, it follows, provided n 1, that Z Z Z Z s p.u, s/dsdu D jsjd p u, dudt C O.1/. Rd jsjn1=d Rd 1jsjn1=d jsj

Observe that s Djsj! with ! 2 Sd1 and ds Djsjd1djsjd!. Therefore, Z Z Z Z Z n1=d p.u, s/dsdu D p.u, !/d!du jsj1djsjCO.1/ d =d d d R jsjn1 RZ S Z 1 1 1 D p.u, !/d!du log.n/ C O.1/. (11.12) d Rd Sd 1

The co-efﬁcient of log.n/ is the noncommutative residue. Hence Res.A/ D ResW .A/.

The residue of a pseudo-differential operator is not always a scalar.

Proposition 11.3.22. There is a compactly supported pseudo-differential operator 1 Rd 1 Rd Q : Cc . / ! Cc . / of order d such that 1 1 Res.Q/ D sin.log.log..1 C n/1=d /// . d nD0

Proof.Letq 2 S d be the symbol in Example 10.2.10, that is,

q.s/ Djsjd .sin C cos/.log.log.jsj///, s 2 Rd , jsj4.

0 1 Rd Let Q be the pseudo-differential operator associated to q.Let 2 Cc . / be such 0 that kk2 D 1. The operator Q D MQ M is compactly supported and Laplacian modulated. Its principal symbol is (see Example 10.2.13)

.u, s/ !j.u/j2 q.s/, u, s 2 Rd .

Using Lemma 11.3.20, and provided n 4d , Z Z Z Z j.u/j2 q.s/dsdu D j.u/j2 du q.s/ds C O.1/ Rd jsjn1=d Rd 4jsjn1=d Z n1=d D q.jsj/jsjd1 djsjCO.1/ 4 D log.n/ sin.log.log.n1=d /// C O.1/.

We prove one more result, however it will not be used until the section on closed Riemannian manifolds. The next lemma implies that the residue can be deﬁned using any equivalent metric on Rd . Section 11.4 Eigenvalues of Laplacian Modulated Operators 355

d d Lemma 11.3.23. Let T : L2.R / ! L2.R / be a compactly based Laplacian modu- Rd Rd lated operator with symbol pT and let A : ! Md . / be a positive matrix valued function such that ajsjjA.u/sjbjsj for all u, s 2 Rd for some 0

pT .u, s/dsdu pT .u, s/dsdu D O.1/, n 0. Rd jsjn1=d Rd jA.u/sjn1=d (11.13)

Proof. For n 0, the conditions on A ensure that bjsjn implies jA.u/sjn and jA.u/sjn implies ajsjn. Therefore, ˇ Z Z Z Z ˇ ˇ ˇ ˇ ˇ ˇ pT .u, s/dsdu pT .u, s/dsduˇ Rd ajsjn ZRd ZjA.u/sjn

jpT .u, s/jdsdu. Rd ajsjnbjsj

The rest of the proof follows in an identical manner to that of Proposition 11.3.18.

11.4 Eigenvalues of Laplacian Modulated Operators

The following eigenvalue theorem is at the heart of Theorem 11.1.1, and hence it is at the heart of Connes’ Trace Theorem. It associates the residue of a compactly supported Laplacian modulated operator to partial sums of eigenvalues of the operator. Consequently, it provides a new estimate for partial sums of eigenvalues of compactly supported pseudo-differential operators. Using the following estimate, the Lidskii formula for a Dixmier trace (see Theorem 10.1.2 or Theorem 7.3.1) will link Dixmier traces to the residue deﬁned in (11.11).

d d Theorem 11.4.1. Let T : L2.R / ! L2.R / be a compactly supported Laplacian modulated operator with symbol pT .ThenT 2 L1,1 and we have

Xn Z Z .j , T/ pT .u, s/dsdu D O.1/, n 0, (11.14) Rd j j 1=d j D0 s n where .T / is an eigenvalue sequence of T (a sequence of non-zero eigenvalues of T in any order such that their absolute values are decreasing, or the zero sequence if T is quasi-nilpotent, see Deﬁnition 1.1.10). The rest of this section proves Theorem 11.4.1. The proof fundamentally relies on the compact support of the Laplacian modulated operator, so that we can use the Dirichlet Laplacian on a bounded domain (which has compact resolvent while the Laplacian on Rd does not) and apply Theorem 11.2.3. 356 Chapter 11 Trace Theorems

d d=2 d d Let em, m 2 Z , be the eigenbasis of .1 / : L2.Œ0, 1 / ! L2.Œ0, 1 / 2ihm,ui d where is the Dirichlet Laplacian. That is, em.u/ :D e , u 2 Œ0, 1 .We d extend em as a multi-periodic function on R .

d d Lemma 11.4.2. Let T : L2.R / ! L2.R / be a Laplacian modulated operator. For every Schwartz function 2 S.Rd /, we have X 2 d kT.em/k D O.t /, t>0. jmj>t

d d Proof.Since.F.em//.s/ D .F/.s m/, s 2 R , m 2 Z , it follows from (11.7) and the Cauchy–Schwartz inequality that Z Z 2 2 j.T .em//.u/j jpT .u, s/j j.F/.s m/jds j.F/.s m/jds Rd Z Rd 2 D const jpT .u, s/j j.F/.s m/jds. Rd We have used the fact that F is a Schwartz function, so it is integrable. Moreover, j.F/.s/jconst hsi2d .Ifjv mj1=2, then

j.F/.s m/jconst hs mi2d const hs vi2d , s 2 Rd and, therefore, Z 2 2 2d j.T .em//.u/j const jpT .u, s/j jhs vi ds. (11.15) Rd

d Deﬁne functions f , h 2 Lmod;1.R / by setting Z 2 2d d f.s/ :D jpT .u, s/j du, h.s/ :Dhsi , s 2 R . Rd For every v with jv mj1=2, it follows from (11.15) that

2 kT.em/k const .f ? h/.v/.

Therefore, X X Z Z 2 kT.em/k const .f ? h/dv const .f ? h/dv. jvmj1=2 jvjt1=2 jmj>t jmjt

It follows from Lemma 11.3.2 that f?h2 Lmod;1. Hence, X 2 d kT.em/k const kf?hkLmod;1O.t /. jmj>t Section 11.4 Eigenvalues of Laplacian Modulated Operators 357

The previous lemma allows us to prove that a compactly supported Laplacian modulated operator is Dirichlet Laplacian modulated. Since the Dirichlet Laplacian has d=2 compact resolvent, and by the Weyl law [38], .1 / 2 L1,1, the following result allows us to use Theorem 11.2.3.

d d Theorem 11.4.3. Let T : L2.R / ! L2.R / be a Laplacian modulated operator. If d d d T is compactly supported in the cube Œ0, 1 , then T : L2.Œ0, 1 / ! L2.Œ0, 1 / is Dirichlet Laplacian modulated.

d Proof. Let em, m 2 Z , be the eigenbasis of the Dirichlet Laplacian indicated above, extended as a multi-periodic function on Rd .Let : Rd ! R be a positive Schwartz function such that D 1onŒ0, 1d .SinceT is compactly supported in Œ0, 1d ,it follows that Tem D T.em/. d Therefore, on the Hilbert space L2.Œ0, 1 /,fort>0wehave 2 X X 1 2 2 TE d=2 0, D kTemk kTemk .1/ t 2 2 2 2=d 1=d 1C4Xjmj t jmj.t=4/ 2 1 D kT.em/k D O.t /. jmj.t=4/1=d

The assertion now follows from Lemma 11.2.5.

Before we can use Theorem 11.2.3 though, we need some technical estimates that will allow us to control the difference Z Z X pT .u, s/duds hTem, emi, t>0. jsjt Rd jmjt

The next two lemmas are purely technical, and the above difference is demonstrated to be bounded in Lemma 11.4.6.

Lemma 11.4.4. Let be a Schwartz function such that D 1 in Œ0, 1d . We have X 2ihu,smi d d e .F/.s m/ D Œ0,t.jsj/ C O.ht jsji /, t>0, s 2 R , jmjt uniformly over u 2 Œ0, 1d .

Proof. Observe that F is a Schwartz function and, therefore, jFj.s/ D O.hsi2d /, s 2 Rd .Forjsjt,wehave ˇ ˇ ˇ X ˇ X ˇ 2 h , i ˇ 2 ˇ e i u s m .F/.s m/ˇ const hm si d const ht jsji d . jmjt jmjt 358 Chapter 11 Trace Theorems

Since is Schwartz, for u 2 Œ0, 1d , X Z X e2ihu,smi.F/.s m/ D e2ihut,si e2ihtu,mi .t/dt Rd m2Zd m2Zd D .u/.

Therefore, X e2ihu,smi.F/.s m/ D .u/ D 1, u 2 Œ0, 1d . m2Zd

Hence, for jsjt and u 2 Œ0, 1d ,wehave ˇ ˇ ˇ X ˇ X ˇ 2 h , i ˇ 2 ˇ e i u s m .F/.s m/ 1ˇ const hm si d jmjt jmjt const ht jsjid .

d d Lemma 11.4.5. Let T : L2.R / ! L2.R / be a compactly based Laplacian modulated operator. If pT is the symbol of T then Z Z d jpT .u, s/jht jsji duds D O.1/, t>0. Rd Rd

Proof. For jsjt=2, we have ht jsjid 2d t d .Forjsj > 2t,wehaveht d d d jsji 2 jsj . Recall that pT .u, s/ D 0whenu is outside of some compact set K.Wehave Z Z Z Z d d d jpT .u, s/jht jsji dsdu 2 t jpT .u, s/jdsdu Rd Rd Z Z ZK Zjsjt=2 d d C jpT .u, s/jdsdu C 2 jsj jpT .u, s/jdsdu. K t=2jsj2t K jsj2t

The ﬁrst term is O.td log.t// by Lemma 11.3.18. The second term is O.1/ by exactly the same argument as in the proof of Lemma 11.3.18. The third term is bounded by the Cauchy–Schwartz inequality.

d d Lemma 11.4.6. Let T : L2.R / ! L2.R / be a Laplacian modulated operator d compactly supported in Œ0, 1 . If pT is the symbol of T then Z Z X pT .u, s/duds hTem, emiDO.1/, t>0. (11.16) jsjt Rd jmjt Section 11.5 Trace Theorem on Rd 359

d Proof. Let be a Schwartz function such that D 1inŒ0, 1 .WehaveTem D T.em/. Observe that .F.em//.s/ D .F/.s m/. It follows from (11.7) that Z Z 2ihu,smi hTem, emiD e pT .u, s/.F/.s m/dsdu. Rd Rd

d Noting that pT .u, s/ D 0foru … Œ0, 1 , we obtain (using Lemma 11.4.4) ˇ Z Z ˇ ˇ X ˇ ˇ ˇ ˇ pT .u, s/duds hTem, emiˇ jsjt Rd ˇ Z Z jmj t ˇ ˇ X ˇ ˇ 2ihu,smi ˇ D ˇ pT .u, s/ e .F/.s m/ Œ0,t.jsj/ dsduˇ Œ0,1d Rd Z Z jmjt d const jpT .u, s/jht jsji dsdu. Œ0,1d Rd

The assertion now follows from Lemma 11.4.5.

We now have the estimates required to prove Theorem 11.4.1.

Proof of Theorem 11.4.1. Without loss we can assume that T is compactly supported d d=2 in Œ0, 1 . Observe that V :D .1 / 2 L1,1,where is the Dirichlet Lapla- cian, is strictly positive, and from Theorem 11.4.3 the operator T is V -modulated. d Theorem 11.2.3 then indicates that T 2 L1,1.L2.Œ0, 1 //. Observe that T D PTP d d d where P projects L2.R / ! L2.Œ0, 1 /, hence, T 2 L1,1.L2.R // since .T / D d .PTP/ D .T jL2.Œ0,1 //. Zd Let vk, k 0, be a rearrangement of the sequence em, m 2 , according to 1=d increasing jmj.Foragivenn 0, let vn D emn with jmnjn . Theorem 11.2.3 indicates that Xn Xn X .j , T/ D hTvj , vj iCO.1/ D hTem, emiCO.1/. j D0 j D0 jmjn1=d

The estimate (11.14) now follows from Lemma 11.4.6.

11.5 Trace Theorem on Rd

We prove Theorem 11.1.1, restated as Theorem 11.5.1 below, which associates the noncommutative residue on Laplacian modulated operators with singular traces. Recall from Section 10.1 the deﬁnition of the weak-l1 ideal of compact operators,

1 L1,1 :DfA 2 L.H / : .n, A/ D O..1 C n/ /g, 360 Chapter 11 Trace Theorems and the Lidskii formula for a Dixmier trace, 1 Xn 1 Tr .A/ D ! .j , A/ , A 2 L 1, ! log.2 C n/ 1, j D0 nD0 where ! is an extended limit on l1 (recall from Section 9.7 that, restricted to L1,1, any extended limit can be used to deﬁne a Dixmier trace) and .A/ 2 l1,1 is an eigenvalue sequence of A. The operator A is said to be Dixmier measurable if Tr!.A/ has the same value for every extended limit !. All Dixmier traces are normalized, meaning that 1 1 Tr! diag D 1. n C 1 nD0

We note that an extended limit ! on l1 is a state on the quotient l1=c0,

1 1 !.Œcn/ :D !.fcngnD0/, fcngnD0 2 l1.

d d Theorem 11.5.1 (Trace Theorem). Suppose T : L2.R / ! L2.R / is a compactly supported Laplacian modulated operator with symbol pT , and Z Z 1 1 Res.T / D pT .u, s/duds , log.2 C n/ jsjn1=d Rd nD0 where Œ denotes the equivalence class in l1=c0 (see Deﬁnition 11.3.19). Then T 2 L1,1 and

(a) Tr!.T / D !.Res.T //

for every extended limit ! on l1.

(b) T is Dixmier measurable if and only if Res.T / is scalar valued and then

Tr!.T / D Res.T /

for every extended limit ! on l1. (c) '.T / D Res.T /

for every normalized trace ' on L1,1 if and only if Res.T / is scalar valued and Z Z

pT .u, s/dsdu D Res.T / log.n/ C O.1/, n 1. (11.17) Rd jsjn1=d Section 11.5 Trace Theorem on Rd 361

Proof. By Theorem 11.4.1 and Deﬁnition 11.3.19, we have that 1 Xn 1 Res.T / D .j , T/ . (11.18) log.2 C n/ j D0 nD0

(a) By Theorem 11.4.1, we have T 2 L1,1. We apply Theorem 10.1.2 and obtain that 1 Xn 1 Tr .T / D ! .j , T/ D !.Res.T //. ! log.2 C n/ j D0 nD0

(b) It follows from (11.18) and Theorem 10.1.3 that T is measurable if and only Res.T / scalar.

1 1 (c) Let D :D diag.f.1Cj/ gj D0/. Note that '.T / D Res.T / (with scalar Res.T /) for every normalized trace ' if and only if T Res.T /D 2 Com.L1,1/.By Theorem 5.7.6 and Proposition 5.7.5, this occurs if and only if

Xn Xn 1 .j , T/ Res.T / D O.1/. (11.19) j C 1 j D0 j D0

By Theorem 11.4.1, this is possible if and only if (11.17) holds. We apply Theorem 11.5.1 to compactly based pseudo-differential operators.

1 Rd 1 Rd Theorem 11.5.2. If A : Cc . / ! Cc . / is a compactly based pseudo-differential operator of order d, then the extension A 2 L1,1 and

Tr!.A/ D !.Res.A// for every extended limit ! on l1.

d d Proof. It follows from Proposition 11.3.17 that A : L2.R / ! L2.R / is Laplacian modulated. There exists a compactly supported function such that A D MA.The 0 0 operator A D AM is compactly supported and the operator A A is compactly 0 0 based and of order 1.ThenRes.AA / D 0 by Lemma 11.3.20. Also AA 2 L1 0 by Corollary 10.2.21. It follows from Theorem 11.5.1 that A 2 L1,1 and, therefore, A 2 L1,1. It now follows from Theorem 11.5.2 that

0 0 Tr!.A/ D Tr!.A / D !.Res.A // D !.Res.A//.

Since Res.A/ is computable from the symbol of A, Theorem 11.5.2 means that we can compute the Dixmier trace of every compactly based pseudo-differential operator, 362 Chapter 11 Trace Theorems including those that are non-classical. Let us indicate that there are pseudo-differential operators whose Dixmier trace depends on the extended limit !, that is, they are not measurable in the sense of Connes.

Corollary 11.5.3. There is a (non-classical) compactly based pseudo-differential op- 1 Rd 1 Rd erator Q : Cc . / ! Cc . / of order d such that the value Tr!.Q/ depends on the extended limit !.

Proof. From Proposition 11.3.22, there exists a compactly based pseudo-differential operator Q of order d such that 1 1 Res.Q/ D sin.log.log.n1=d /// . d nD0

Clearly, Res.Q/ is not a scalar and, thus, Tr!.Q/ depends on the extended limit !.

The situation is far different for classical pseudo-differential operators. The corollary below was stated by Connes in [45] for Dixmier traces, however, every normalized trace on L1,1 computes the scalar noncommutative residue.

1 Rd 1 Rd Corollary 11.5.4 (Connes’ Trace Theorem). Suppose A : Cc . / ! Cc . / is a classical compactly based pseudo-differential operator of order d. Then the extension A 2 L1,1 and, for every normalized trace ' on L1,1, we have

'.A/ D ResW .A/.

Here ResW denotes the noncommutative residue of Wodzicki as described in Sec- tion 10.2.

Proof. By using the same trick as in the proof of Theorem 11.5.2 we may take A, without loss, as compactly supported. From Proposition 11.3.21, Res.A/ D ResW .A/ is a constant and from the proof of that proposition, equation (11.12) in particular, we obtain that (11.17) is satisﬁed. The assertion now follows from Theorem 11.5.1.

11.6 Trace Theorem on Closed Riemannian Manifolds

We now transfer the notion of Laplacian modulated operators and their results to closed Riemannian manifolds. A Hodge–Laplacian modulated operator on a closed manifold, as we define below, is locally a compactly supported Laplacian modulated operator, and the converse is true. We define the residue using the eigenvectors of the Hodge–Laplacian. By showing that Hodge–Laplacian operators are locally Laplacian modulated operators we will use the symbols of Laplacian modulated operators to associate the residue defined using eigenvectors to the integral of symbols over co-disc Section 11.6 Trace Theorem on Closed Riemannian Manifolds 363 bundles. In this way we can derive Connes’ Trace Theorem and a purely spectral formula for the noncommutative residue of Wodzicki on a closed Riemannian manifold.

Hodge–Laplacian Modulated Operators In this section we introduce Hodge–Laplacian modulated operators and their vector- valued residue. Suppose X is a d-dimensional closed (and for the purpose of convenient proofs, orientable) Riemannian manifold. From the results in Section 10.3 the Laplace–Beltrami operator associated to a Riemannian metric g admits a unique extension as a negative unbounded operator 2 g : H .X/ ! L2.X/ and d=2 .1 g / : L2.X/ ! L2.X/ is a strictly positive compact operator (alternatively, the Laplace–Beltrami operator g has compact resolvent). As a consequence, the Laplace–Beltrami operator on a closed manifold X has crucial spectral properties denied to the Laplacian on Euclidean space. 1 First, there exists an orthonormal basis fengnD0 of eigenvectors of the Hodge– Laplacian, 2 g en D nen, n 0,

2 2 ordered such that the eigenvalues 0 1 are increasing. For this section 1 d=2 fengnD0 denotes this ordered eigenbasis of .1 g / . Second, by Weyl’s asymptotic formula [38], d 1 hni Cd .1 C n/ for a constant Cd . Therefore, d=2 .1 g / 2 L1,1 belongs to the weak-l1 ideal. These properties are the foundation of spectral geometry [38], and they have markedly strong consequences for the study of the noncommutative residue.

Deﬁnition 11.6.1. A bounded operator T : L2.X/ ! L2.X/ is Hodge–Laplacian d=2 modulated if it is .1 g / -modulated, that is, if, for some Riemannian metric g, we have 1=2 d=2 1 sup t kT.1 C t.1 g / / k2 < 1, t>0 where kk2 denotes the Hilbert–Schmidt norm on L2.L2.X//. 364 Chapter 11 Trace Theorems

By the following lemma, Deﬁnition 11.6.1 does not depend on the particular choice of Riemannian metric g.

Lemma 11.6.2. If T : L2.X/ ! L2.X/ is as in Deﬁnition 11.6.1,thenT is .1 d=2 g/ -modulated for every Riemannian metric g.

Proof. If g1 and g2 are Laplace–Beltrami operators on X with respect to different d=2 d=2 Riemannian metrics, then .1g1 / const.1g2 / . The assertion follows from Remark 11.2.7.

A pseudo-differential operator of order d is Hodge–Laplacian modulated.

Proposition 11.6.3. If A : C 1.X/ ! C 1.X/ is a pseudo-differential operator of order d then (the extension of) A is Hodge–Laplacian modulated.

d=2 1 Proof. For brevity, set V :D .1 g / . The operator B :D AV is a pseudo- differential operator of order 0 and is, therefore, bounded. The operator V belongs to L1,1 and, by Lemma 11.2.8, is Hodge–Laplacian modulated. So then is A D BV by the fact that the set of modulated operators form a left ideal.

The Banach space of Hodge–Laplacian modulated operators is a bimodule for the pseudo-differential operators of order 0.

Lemma 11.6.4. If T : L2.X/ ! L2.X/ is a Hodge–Laplacian modulated operator and if A is (the extension of) a pseudo-differential operator of order 0, then both TA and AT are Hodge–Laplacian modulated.

Proof. We need only show that TA is Hodge–Laplacian modulated since A has a bounded extension, and the set of Hodge–Laplacian modulated operators is a left ideal d=2 of L.L2.X//. For brevity, set V :D .1 g / . Observe that ŒV , A is a pseudo- differential operator of order d 1 and, therefore, ŒV , A D BV with B a bounded operator. We have, for t>0,

tV ŒA, .1 C tV /1 D t.1 C tV /1ŒV , A.1 C tV /1 D .1 C tV /1 B . 1 C tV Hence, 1 1 tV 1=2 kTŒA, .1 C tV / k2 kT.1 C tV / k2 kAk1 D O.t / 1 C tV 1 and, therefore,

1 1 1 1=2 kTA.1 C tV / k2 kTŒA, .1 C tV / k2 CkT.1 C tV / Ak2 D O.t /. Section 11.6 Trace Theorem on Closed Riemannian Manifolds 365

From the spectral properties of the Laplace–Beltrami operator we have that .1 d=2 d=2 g/ 2 L1,1. Therefore, unlike the similar operator .1 / on Euclidean space, we can invoke Theorem 11.2.3 directly in order to calculate traces of Hodge- Laplacian modulated operators. The difﬁculty resides in the appropriate notion of a residue of a Hodge-Laplacian modulated operator. We deﬁne the residue using the eigenvectors of the Hodge–Laplacian.

1 Lemma 11.6.5. If fengnD0 is an ordered eigenbasis of the Hodge–Laplacian and T is a Hodge–Laplacian modulated operator, then

1 Xn hTe , e iDO.1/, n 0. log.2 C n/ j j j D0

Proof. From Theorem 11.2.3 we have that T 2 L1,1. Hence, .n, T/ D O..1 C n/1/, n 0, and, combining with Theorem 11.2.3(b), ˇ ˇ ˇ ˇ ˇ Xn ˇ ˇ Xn ˇ ˇ ˇ ˇ ˇ ˇ hTej , ej iˇ ˇ .j , T/ˇ C O.1/ j D0 j D0 Xn .j , T/C O.1/ D O.log.2 C n//. j D0

The lemma shows that the following function is well deﬁned.

Deﬁnition 11.6.6. The vector-valued linear function 1 Xn 1 Res.T / :D hTe , e i , log.2 C n/ j j j D0 nD0

1 where Œ denotes the equivalence class in `1=c0 and fengnD0 is an ordered eigenbasis of the Hodge–Laplacian, is called the residue of the Hodge–Laplacian modulated operator T . The next section shows that the residue is a considerable extension of the scalar noncommutative residue on classical pseudo-differential operators of order d.

Symbols of Hodge–Laplacian Modulated Operators Hodge–Laplacian modulated operators are locally Laplacian modulated operators.

Proposition 11.6.7. Let X be a closed d-dimensional Riemannian manifold and let .U , h/ be a chart. If T : L2.X/ ! L2.X/ is a Hodge–Laplacian modulated operator 366 Chapter 11 Trace Theorems

1 Rd Rd that is compactly supported in U , then Wh TWh : L2. / ! L2. / is a compactly supported Laplacian modulated where Wh : L2.U / ! L2.h.U //. d=2 1 Proof. For brevity, set V :D .1g / .Let 2 C .X/ be a function compactly supported in U such that M TM D T . By Lemma 11.2.6, T D TM is jVMj- modulated. The latter operator is compactly supported in U . 2 1 2 The positive pseudo-differential operator D D Wh jVM j Wh is compactly supported and p p 1 Rd Rd Wh TWh : L2. , detgdm/ ! L2. , detgdm/ is D-modulated. Here det g is the determinant of the coordinates of the metric in the chart U , as explained in Section 10.3. Sincep the respective norms of a compactly sup- d d ported function are equivalent in L2.R , detgdm/ and in L2.R /, it follows that 1 Rd Rd Wh TWh : L2. / ! L2. / is D-modulated. Since D2 is a pseudo-differential operator of order 2d, there exists a pseudo- differential operator P of order 0 such that

2 d=2 d=2 D D .1 g / P.1 g/ .

Since P is order 0, then kP k1 < 1 and

2 d D kP k1 .1 / .

It follows that D const .1 /d=2 by preservation of order by the square root operation on positive operators. The assertion follows by applying Remark 11.2.7.

To associate a Hodge–Laplacian modulated operator to a function on the tangent bundle (a symbol) we emulate the usual treatment of pseudo-differential operators. The symbol is deﬁned locally by the restriction to a chart, pulled back to the tangent bundle and then patched together using a partition of unity. We extend Theorem 11.4.1 to a (chart dependent) statement involving the tangent bundle.

Lemma 11.6.8. Let T : L2.X/ ! L2.X/ be Hodge–Laplacian modulated operator. If .U , h/ is a chart and if T is compactly supported in U , then

Xn Z Z .k, T/ p 1 .x, s/dxds D O.1/, n 0, Wh TWh jsjn1=d Rd kD0 where .T / is an eigenvalue sequence of T .

1 Proof. By Proposition 11.6.7, Wh TWh is Laplacian modulated and it is clear that it is compactly supported. The assertion now follows from Theorem 11.4.1 since eigenvalues are invariant under the isometry Wh : L2.h.U // ! L2.U /. Section 11.6 Trace Theorem on Closed Riemannian Manifolds 367

Let .Ui , hi /,1 i N , be an atlas of the manifold X. Take a smooth partition of unity j ,1 j M , such that for every pair .j , k/ with j k ¤ 0 both j and k are compactly supported in some Ui . We will always use such a partition of unity M and we label it by ‰ Df j gj D1. With respect to such a partition of unity we are able to deﬁne a coordinate dependent symbol.

Deﬁnition 11.6.9. If T : L2.X/ ! L2.X/ is a Hilbert–Schmidt operator, then deﬁne .‰,g/ Rd a coordinate dependent symbol pT 2 L2.X / by the formula

XM q .‰,g/ 1 1 p .u, s/ :D Ui .u/ det.g /.hi .u//pW M TM Wh .hi .u/, s/, T hi j k i j ,kD1 where i is chosen so that the support of j and k belong to Ui .

Deﬁnition 11.6.10. An operator T : L2.X/ ! L2.X/ is called localized if 1 MTM 2 L1 for every disjointly supported , 2 C .X/.

For a given partition ‰ of unity, an operator is called ‰-localized if M j TM k 2 L1 for every pair .j , k/ with j k D 0.

Theorem 11.6.11. Let .X, g/ be a closed d-dimensional Riemannian manifold. Sup- pose that the Hodge–Laplacian modulated operator T : L2.X/ ! L2.X/ is ‰- localized. We have Xn Z Z .‰,g/ .m, T/ pT .u, s/duds D O.1/, n 0, j j 1=d mD0 s n X where .T / is an eigenvalue sequence of T .

Proof. Every operator M j TM k is Hodge–Laplacian modulated by Lemma 11.6.4.

If j k ¤ 0, then M j TM k is compactly supported in some Ui . By Proposi- tion 5.7.5, we have

Xn XM Xn

.m, T/D .m, M j TM k / C O.1/. mD0 j ,kD1 mD0

The assertion now follows from Lemma 11.6.8.

We deﬁne an equivalence class of symbols and show that, for localized Hodge– Laplacian modulated operators, the residue is deﬁned by the equivalence class. The next result says that every localized Hodge–Laplacian modulated operator can be as- signed a coordinate and metric independent “principal” symbol. 368 Chapter 11 Trace Theorems

Corollary 11.6.12. Let X be a d-dimensional closed Riemannian manifold and let g1, g2 be Riemannian metrics on X. If T is a localized Hodge–Laplacian modulated operator, then Z Z Z Z .‰1,g1/ .‰2,g2/ pT .u, s/duds pT .u, s/duds D O.1/ jsjn1=d X jsjn1=d X and Z Z Z Z p.‰1,g1/.u, s/duds p.‰1,g1/.u, s/duds D O.1/ T = T 1=d 1 2 1=d jsjn X X jg1 .u/sjn for partitions ‰1, ‰2 of unity.

Proof. The ﬁrst assertion follows from Theorem 11.6.11 since eigenvalues are metric and coordinate independent. The second assertion follows from Lemma 11.3.23.

The differences in the lemma deﬁne equivalence relations on coordinate dependent symbols. That is, two coordinate dependent symbols are equivalent if they satisfy the displays in Corollary 11.6.12. We will only be concerned with coordinate dependent symbols up to this equivalence.

Deﬁnition 11.6.13. If T : L2.X/ ! L2.X/ is a localized Hodge–Laplacian modu- .‰,g/ lated operator, then the equivalence class ŒpT of any coordinatedependent symbol .‰,g/ pT is called the symbol of T (that all coordinate dependent symbols generate the same class is shown in Corollary 11.6.12 above).

Wemayfixametricg and we may take any coordinate dependent symbol to act as a representative for the symbol when discussing localized Hodge–Laplacian modulated .‰,g/ operators. Hence, we let pT denote a coordinate dependent symbol pT , dropping explicit reference to the coordinates and the metric. For a localized Hodge–Laplacian modulated operator we characterize its residue (as in Definition 11.6.6, which was a purely spectral definition) in terms of its symbol.

Theorem 11.6.14. Let T : L2.X/ ! L2.X/ be a localized Hodge–Laplacian modulated operator. Then the residue of T depends only on its symbol pT , and Z Z 1 1 Res.T / D pT .u, s/duds (11.20) =d log.2 C n/ jsjn1 X nD0 where Œ denotes the equivalence class in l1=c0. Section 11.6 Trace Theorem on Closed Riemannian Manifolds 369

1 Proof. If fej gj D0 is an ordered eigenbasis for the Hodge–Laplacian, then Theo- rem 11.2.3 proves that

Xn Xn hTej , ej iD .j , T/C O.1/. j D0 j D0

If pT is any coordinate dependent symbol in the equivalence class associated to T , then Theorem 11.6.11 and Corollary 11.6.12 prove that

Xn Z Z .j , T/D pT .u, s/duds C O.1/. j j 1=d j D0 s n X

The assertion follows from Deﬁnition 11.6.6.

It also follows that, if pT is well deﬁned as a function on the tangent bundle T X, then equivalently the residue is deﬁned in terms of co-disc bundles " Z # 1 1 Res.T / D pT .v/dv (11.21) =d log.2 C n/ D .n1 /X nD0 where dv is the volume form on T X. Pseudo-differential operators on manifolds are localized operators. We can now show that the residue of a classical pseudo-differential operator of order d is Wodzicki’s noncommutative residue.

Corollary 11.6.15. If A : C 1.X/ ! C 1.X/ is a classical pseudo-differential operator of order d, then the extension A is a localized Hodge–Laplacian modulated operator with

Res.A/ D ResW .A/ where ResW is Wodzicki’s noncommutative residue (as in Section 10.3). Proof. That A is localized follows from the definition of a pseudo-differential operator (see Definition 10.3.2). Suppose A is also of order m<d.ThenA is trace class and, by Definition 11.6.6, Res.A/ D 0. Thus, the residue depends only on the principal symbol pd of A. It follows from Theorem 11.6.14 that Z Z 1 1 Res.A/ D pd .u, s/duds . =d log.2 C n/ 1jsjn1 X nD0 370 Chapter 11 Trace Theorems

d1 d Setting s Djsj! with ! 2 S ,wehavepd .u, s/ Djsj pd .u, !/ if jsj1. Therefore, Z Z

pd .u, s/duds 1jsjn1=d X Z Z Z n1=d d d1 D jsj pd .u, !/jsj djsjd! du XZ ZSd 1 1 1 D pd .u, !/d!du log.n/ C O.1/. (11.22) d X Sd 1 Hence, Z Z Z 1 1 Res.A/ D pd .u, !/d!du D pd .s/ds, d X Sd 1 d SX where ds is the volume form on SX. The last equality in the display follows by previous remarks as the principal symbol of A is well deﬁned on the tangent bundle, [221].

Let us state what this means as an explicit spectral formula for the noncommutative residue. The next assertion follows immediately from Corollary 11.6.15 and the deﬁnition of the residue on Hodge–Laplacian modulated operators, Deﬁnition 11.6.6.

Corollary 11.6.16 (Spectral formula for the noncommutative residue). Let A : C 1.X/ ! C 1.X/ be a classical pseudo-differential operator of order d, 1 and let fengnD0 be an ordered eigenbasis of the Hodge–Laplacian g .Then

1 Xn ResW .A/ D lim hAej , ej i n!1 log.2 C n/ j D0 where ResW is Wodzicki’s noncommutative residue.

Traces of Hodge–Laplacian Modulated Operators We use Corollary 11.2.4 to identify traces of Hodge–Laplacian modulated operators with their residue.

Theorem 11.6.17 (Trace Theorem on manifolds). Let T : L2.X/ ! L2.X/ be a Hodge–Laplacian modulated operator. Then T 2 L1,1 and (a)

Tr!.T / D !.Res.T //

for every extended limit ! on l1. Section 11.6 Trace Theorem on Closed Riemannian Manifolds 371

(b) T is Dixmier measurable if and only if Res.T / is scalar valued and then

Tr!.T / D Res.T /

for every extended limit ! on l1. (c) '.T / D Res.T /

for every normalized trace ' on L1,1 if and only if Res.T / is scalar valued and satisﬁes the condition Xn hTej , ej iDRes.T / log.n/ C O.1/, n 1, j D0 (or Z Z

pT .u, s/duds D Res.T / log.n/ C O.1/, n 1, (11.23) jsjn1=d X

if T is localized and pT is the symbol of T .) Proof. All statements follow immediately from the deﬁnition of the residue, Corol- lary 11.2.4, and Theorem 11.6.14.

The ﬁrst corollary of Theorem 11.6.17 is that the Dixmier trace of any pseudo- differential operator of order d can be computed from local symbols.

Corollary 11.6.18. Let A : C 1.X/ ! C 1.X/ be a pseudo-differential operator of order d. Then the extension A 2 L1,1 is Hodge–Laplacian modulated and localized and

Tr!.A/ D !.Res.A// for every extended limit ! on l1. The second corollary is that there are non-measurable pseudo-differential operators of order d. Hence, the set of pseudo-differential operators of order d does not have a unique trace.

Corollary 11.6.19. There exists a (non-classical) pseudo-differential operator Q : 1 1 C .X/ ! C .X/ of order d such that the value Tr!.Q/ depends on the extended limit !.

Proof. Select a chart .U , h/ and let the operator Q be compactly supported in U .In local coordinates, set Q to be the operator constructed in Corollary 11.5.3. 372 Chapter 11 Trace Theorems

The third corollary of Theorem 11.6.17 is Connes’ Trace Theorem for manifolds. The previous corollary shows that the qualiﬁer classical cannot be omitted from the following statement.

Corollary 11.6.20 (Connes’ Trace Theorem). Let A : C 1.X/ ! C 1.X/ be a classical pseudo-differential operator of order d. We have A 2 L1,1 and

'.A/ D ResW .A/ for every normalized trace ' on L1,1 where ResW is Wodzicki’s noncommutative residue.

Proof. The assertion follows from Theorem 11.6.17(c) as, from the proof of Corol- lary 11.6.15 (in particular, the equation (11.22)), we have that Z Z

p.u, s/duds D ResW .A/ log.n/ C O.1/, n 1, jsjn1=d X where p is the symbol (or equivalently, principal symbol) of the operator A.

11.7 Integration of Functions

Laplacian modulated operators are a wider class than pseudo-differential operators. d=2 d d We show in this section that the operator Mf .1 / : L2.R / ! L2.R /, explicitly d=2 d x.t/ ! f .t/..1 / x/.t/, x 2 L2.R /, (11.24) is Laplacian modulated for any square integrable function f (here .1 /d=2 is the bounded positive operator described in Example 10.2.16). Theorem 11.5.1 is used to compute the singular trace of the operator (11.24) when f is compactly supported. It turns out that this computes the Lebesgue integral of the function f . For a locally bounded measurable function f on Rd deﬁne an (unbounded) normal operator Mf by setting

Rd .Mf x/.t/ :D f.t/x.t/, x 2 L1. /, x has compact support .

d=2 Lemma 11.7.1. The operator Mf .1 / is Hilbert–Schmidt if and only if f 2 d L2.R /.

d 2 2 d=2 d Proof. First, let f 2 L2.R /.Setg.t/ :D .1 C 4 jtj / , t 2 R .Wehave d=2 1 1 .1 / F D F Mg . Thus, Z d=2 1 2ihu,si .Mf .1 / F x/.u/ D f.u/g.s/e x.s/ds. (11.25) Rd Section 11.7 Integration of Functions 373

d=2 1 Hence, Mf .1 / F is an integral operator with square integrable kernel (that is, a Hilbert–Schmidt operator). Since the Fourier transform is a unitary operation, d=2 then the operator Mf .1 / is Hilbert-Schmidt. d=2 Conversely, let Mf .1 / be a Hilbert–Schmidt operator. Without loss of d generality, f 0. Fix a cube K and let fn :D minfn, fK g2L2.R /, n 0. It follows from (11.25) that

d=2 d=2 const kfnk2 DkMfn .1 / k2 kMf .1 / k2.

Taking the limit n !1, we obtain that

d=2 kfK k2 const kMf .1 / k2.

Since the cube K is arbitrarily large, the assertion follows.

Modulation is Not Preserved by Taking Adjoints The Laplacian modulated operators form a left ideal inside the Hilbert–Schmidt operators. They do not form a two-sided ideal; we use operators of the form (11.24) to give an explicit example where the adjoint of a Laplacian modulated operator is not Laplacian modulated.

Lemma 11.7.2. If g.s/ :D .1 C 42jsj2/d=2, s 2 Rd , then

p d=2 .u, s/ D f.u/g.s/, Mf .1/Z 2ihu,vi 1 d= F p.1/ 2Mf .u, s/ D e g.s v/. f /.v/dv. Rd

d=2 1 Proof.Wehave.1 / D F Mg F and, therefore, Z d=2 2ihu,si F Rd .Mf .1 / x/.u/ D f.u/g.s/e . x/.s/ds, x 2 L2. /, Rd and the ﬁrst assertion follows from the deﬁnition of the symbol. d Similarly, for x 2 L2.R /, Z d=2 2ihu,vi ..1 / Mf x/.u/ D e g.v/.F.f x//.v/dv Z Z Rd D e2ihu,vig.v/.Fx/.s/.Ff /.v s/dvds Z Rd RdZ D e2ihu,si.Fx/.s/ e2ihu,vsig.v/.Ff /.v s/dv ds. Rd Rd 374 Chapter 11 Trace Theorems

Hence, Z 2ihu,vsi d= F p.1/ 2Mf .u, s/ D e g.v/. f /.v s/dv Rd Z D e2ihu,vig.v C s/.Ff /.v/dv Rd Z D e2ihu,vig.s v/.F1f /.v/dv. Rd

The next result provides an example where right multiplication by a square integrable function does not yield a Laplacian modulated operator but left multiplication does. It also provides a condition on the Fourier transform of a square integrable function so that both left and right multiplication yield a Laplacian modulated operator. d d The Banach space Lmod;2.R /, a subset of L2.R /, was introduced in Section 11.3, see Deﬁnition 11.3.6.

Proposition 11.7.3. We have that d=2 d (a) Mf .1 / is Laplacian modulated for every f 2 L2.R /. Rd d=2 (b) there exists f 2 Lmod;2. / such that the operator .1/ Mf is not Lapla- cian modulated. d=2 F Rd (c) .1/ Mf is Laplacian modulated provided that f 2 Lmod;2. /, where F denotes the Fourier transform.

Proof. Deﬁne a function g by setting

g.t/ :D .1 C 42jtj2/d=2, t 2 Rd .

2 d Recall that jgj 2 Lmod;1.R /. d=2 (a) Let p be the symbol of Mf .1 / . It follows from Lemma 11.7.2 that Z 2 2 2 Rd jp.u, /j du Dkf k2jgj 2 Lmod;1. /. Rd

d=2 By Lemma 11.3.13, the operator Mf .1 / is Laplacian modulated.

1 1C2˛=d (b) Let f˛ be the function constructed in Example 11.3.7. We have F f˛ D g 2 d d=2 and jf˛j 2 Lmod;1.R /.Ifp is a symbol of the operator .1 / Mf ,then Section 11.7 Integration of Functions 375

it follows from Lemma 11.7.2 that Z 2 2 1 2 jp.u, /j du Djgj ? jF f˛j Rd 2 2C4˛=d 2C4˛=d d D g ?g const g … Lmod;1.R /.

d=2 By Lemma 11.3.13, the operator .1 / Mf is not Laplacian modulated.

d=2 (c) If p is the symbol of the operator .1 / Mf , then it follows from Lem- ma 11.7.2 and Lemma 11.3.2 that Z 2 2 1 2 d jp.u, /j du Djgj ? jF f j 2 Lmod;1.R /. Rd Hence, the operator is Laplacian modulated by Lemma 11.3.13.

Integration of Compactly Supported Square Integrable Functions on Rd

1 d d=2 If f 2 C .R / is a compactly supported function, then Mf .1 / is a classical pseudo-differential operator of order d. It has been noted by many authors that Connes’ original trace theorem implied Z Sd1 d=2 Vol . / 1 Rd Tr!.Mf .1 / / D f.u/du, f 2 Cc . /. d.2/d Rd R This led a Dixmier trace Tr! to be denoted by a symbol (under the assumption of measurability in the sense of Connes) and termed integration. In fact, from Corol- lary 11.5.4, Z Sd1 d=2 Vol . / 1 Rd '.Mf .1 / / D f.u/du, f 2 Cc . /, d.2/d Rd for every normalized trace ' on L1,1. We improve on the condition that f be smooth. Rd .dC1/=2 Theorem 11.7.4. If f 2 L2. / is compactly supported, then Mf .1/ 2 L1. Proof. The proof of this well-known result [222, Theorem 4.5], is provided, amongst similar results, in Chapter 4 of [222].

d Theorem 11.7.5. If f 2 L2.R / is a compactly supported function, then Mf .1 d=2 / 2 L1,1 and Z Sd1 d=2 Vol . / '.Mf .1 / / D f.u/du d.2/d Rd for every normalized trace ' on L1,1. 376 Chapter 11 Trace Theorems

1 Rd Proof. Let 2 Cc . / be such that f D f . By Proposition 11.7.3, the operator d=2 Mf .1 / is Laplacian modulated. It follows from Lemma 11.6.4 that Mf .1 d=2 / M is a compactly supported Laplacian modulated operator. d=2 By Theorem 11.5.1 we have that Mf .1 / M 2 L1,1. Deﬁne the pseudo- d=2 differential operator Q :D ŒM , .1 / , which, by Example 10.2.13 is of order d 1. Since Q is order d 1 then there exists a zero order pseudo-differential operator R such that Q D .1 /.dC1/=2R.Wehave

d=2 d=2 .dC1/=2 Mf .1 / Mf .1 / M D Mf Q D Mf .1 / R.

.dC1/=2 By Theorem 11.7.4 the operator Mf .1 / is trace class and by Proposi- .dC1/=2 tion 10.2.15 the operator R has a bounded extension. Hence, Mf .1 / R is trace class and

d=2 d=2 Mf .1 / Mf .1 / M 2 L1.

d=2 L L Hence, Mf .1 / 2 1,1 and, for any trace ' on 1,1,

d=2 d=2 '.Mf .1 / / D '.Mf .1 / M/, (11.26)

L d=2 since ' vanishes on 1. Denote by p the symbol of the operator Mf .1/ and by d=2 r the symbol of the operator Mf .1 / M.Then.p r/.u, s/ D f.u/q.u, s/, where q is the symbol of the operator Q. Hence, ˇ Z Z ˇ Z Z ˇ ˇ ˇ ˇ ˇ .p r/.u, s/dsduˇ const jf.u/j.1 Cjsj/ d 1 dsdu D O.1/. Rd jsjt Rd Rd This shows that the symbol r satisﬁes (11.17), since p is easily checked to satisfy (11.17). It also shows that, by the deﬁnition of the vector-valued residue of compactly based Laplacian modulated operators,

d=2 d=2 Res.Mf .1 / / D Res.Mf .1 / M/. (11.27)

d=2 Since r satisﬁes (11.17) and Mf .1 / M is compactly supported then, by Theorem 11.5.1,

d=2 d=2 '.Mf .1 / M / D Res.Mf .1 / M/. (11.28)

The assertion of the theorem now follows from (11.26), (11.27) and (11.28).

Integration of Square Integrable Functions on a Closed Manifold Suppose that X is a d-dimensional closed Riemannian manifold with metric g.If f 2 L2.X/, then deﬁne Mf : L1.X/ ! L2.X/ by Mf x.u/ :D f .u/x.u/, Section 11.7 Integration of Functions 377

x 2 L1.X/. In this section g denotes the Laplace–Beltrami operator (see Deﬁ- d=2 nition 10.3.3) and .1 g/ 2 L1,1 is the compact operator described in Exam- ple 10.3.5 and Section 11.6. We start with the following analog of Theorem 11.7.4.

.dC1/=2 Theorem 11.7.6. If f 2 L2.X/, then Mf .1 g/ 2 L1.L2.X//. Proof. We refer the reader to Chapter 4 of [222].

d=2 Lemma 11.7.7. If f 2 L2.X/, then the operator Mf .1 g/ is localized. Proof. If , 2 C 1.X/ are such that D 0, then

d=2 d=2 M.Mf .1 g / /M D MfŒ.1 g / , M .

d=2 By Example 10.2.13,the operator Œ.1g / , M is a pseudo-differential operator of order d 1. It follows that

d=2 .dC1/=2 M.Mf .1 g/ /M D Mf.1 g / P where P is a pseudo-differential operator of order 0. Since P is bounded, the assertion follows from Theorem 11.7.6.

d=2 Proposition 11.7.8. If f 2 L2.X/, then the operator Mf .1 g/ is Hodge– Laplacian modulated.

Proof. Fix a chart .U , h/. Without loss of generality, f is compactly supported in U . Select 2 C 1.X/ compactly supported in U and such that f D f .Wehave

1 2 2 d= d= Wh .Mf .1 g / M/Wh D Mf ıh 1 .1 / P ,

d=2 where Mf ıh1 .1 / is a Laplacian modulated operator (by Proposition 11.7.3) and 2 1 2 d= d= P D .1 / Wh .1 g/ Wh Mıh 1 is a pseudo-differential operator of order 0. By Lemma 11.6.4,

1 d=2 Wh .Mf .1 g/ M/Wh is a compactly supported Laplacian modulated operator. By Proposition 11.6.7, d=2 Mf .1 g / M is Hodge–Laplacian modulated. 378 Chapter 11 Trace Theorems

d=2 Recall that Œ.1 g / , M1 is a pseudo-differential operator of order d 1 (see Example 10.2.13 for the local result). Deﬁne a pseudo-differential operator P of order 1 by setting

d=2 d=2 Œ.1 g/ , M1 D .1 g / P .

An easy inductive argument shows that

d=2 d=2 dC1 .1 g / D .1 g/ P Xd d=2 k d=2 k C M1.1 g/ P C .1 g / MP . kD0

Since Mf M1 D 0, it follows that

Xd d=2 d=2 k d=2 dC1 Mf .1 g / D .Mf .1 g / M/P C Mf .1 g / P . kD0

d=2 By the previous paragraph, Mf .1 g/ M is Hodge–Laplacian modulated. By d=2 k Lemma 11.6.4, so is .Mf .1 g/ M /P for k 0. Note that

d=2 dC1 .dC1/=2 d=2 Mf .1 g / P D .Mf .1 g / / .1 g/ Q with Q being a pseudo-differential operator of order 0. The operator Mf .1 d=2 dC1 g/ P is hence Hodge–Laplacian modulated by the left ideal property of the set of Hodge–Laplacian modulated operators and by Lemma 11.6.4. Thus, Mf .1 d=2 g/ is Hodge–Laplacian modulated.

Proposition 11.7.9. If f 2 L2.X/, then Z Vol.Sd1/ Res.M .1 /d=2/ D f.u/du. f g d d.2/ X Proof. Fix some chart .U , h/. Without loss of generality, f is compactly supported d=2 in U . Lemma 11.7.7 and Lemma 11.7.8 provide that Mf .1 g / is a localized Hodge–Laplacian modulated operator. Its symbol p (the equivalence class in the sense of Deﬁnition 11.6.13) is generated by the function on U Rd , q p.u, s/ D det.g1/.h.u//f .u/.2/d jg1=2.h.u//sjd , Section 11.7 Integration of Functions 379 see Example 10.3.5, and Z Z p.u, s/duds =d jsjn1 X Z Z 1 D f ı h1.t/jg1=2.t/sjd dsdt C O.1/. d .2/ Rd jsjn1=d

Using the equivalences in Corollary 11.6.12, Z Z 1 D f ı h1.t/jg1=2.t/sjd dsdt C O.1/. d .2/ Rd jg1=2.t/sjn1=d

Using the change of coordinates in s, Z Z 1 p D f ı h1.t/jsjd detg.t/dsdt C O.1/ d d =d .2/ RZ jsjn1 Z 1 p D f ı h1.t/ detg.t/dt jsjd ds C O.1/ .2/d Rd j j 1=d Z s n Vol.Sd1/ D f.u/du log.n/ C O.1/. (11.29) d d.2/ X

Integration of a square integrable function on a closed Riemannian manifold is recovered by any trace.

d=2 L Theorem 11.7.10. If f 2 L2.X/, then Mf .1 g/ 2 1,1 and Z Vol.Sd1/ '.M .1 /d=2/ D f.u/du f g d d.2/ X for every normalized trace ' : L1,1 ! C. Proof. The ﬁrst assertion follows from Proposition 11.7.8 and Lemma 11.2.9. The second assertion follows from Proposition 11.7.9, equation (11.29), and Theo- rem 11.6.17(c).

The condition that f is square integrable cannot be dropped. Adapting the proof of d=2 Lemma 11.7.1 it can be shown that Mf .1 g / is Hilbert–Schmidt if and only d=2 L if f 2 L2.X/. It follows that Mf .1 g/ 2 1,1 if and only if f 2 L2.X/, see also [151]. 380 Chapter 11 Trace Theorems

11.8 Notes

Origin of the Approach Connes’ original trace theorem is usually where the foundation of the noncommutative residue ends and applications begin (the rationale derived from the theorem is that Dixmier traces are the “noncommutative residue” for noncommutative versions of pseudo-differential operators). We thought, however, there was more to the story behind Connes’ Trace Theorem. The authors, with Nigel Kalton and Denis Potapov, questioned whether Dixmier traces were the only singular traces that identify with the noncommutative residue. Results of Wodzicki indicated that they were not. We also questioned whether the Dixmier trace of an arbitrary (a not necessarily classical) pseudo-differential operator of order d had a similar formula. The answers we found were reported in [126], which is the origin of the approach to trace theorems covered in this chapter.

Laplacian Modulated Operators d d A function L2.R R / considered as the symbol of a Hilbert–Schmidt operator is a common view of extending the notion of a pseudo-differential operator [58]. The Banach space of symbols associated to the Banach space of Laplacian modulated operators and the concept of a Laplacian modulated operator were introduced by the authors in [126]. The extension of the noncommutative residue from order d pseudo-differential operators to compactly-based Laplacian modulated operators was developed in [126]. There are various examples of extending the noncommutative residue for speciﬁc pseudo-differential operator classes, e.g. [17]. The advantage of the estimate in the eigenvalue theorem, Theorem 11.4.1, is the ability to calculate traces, as shown by the trace theorem, Theorem 11.5.1. Classical texts on pseudo- differential operators do not usually consider the ideal L1,1 explicitly. It features naturally in the classical approach to spectral asymptotics of strictly positive hypoelliptic operators, see [221, §30 ], but not for non-elliptic operators. Estimates of the singular values of operators of the form Mf g./, are known as Cwikel estimates (see [13,57], [222, Chapter 4], [251]). The estimates for singular values that we know of which are the closest to the modulation condition have appeared in analysis using Gabor frames, see [110]. The trace theorem, Theorem 11.5.1, is an extension of Connes’ Trace Theorem. Connes’ statement involving Dixmier traces ﬁrst appeared in [45, Theorem 1]. Dixmier introduced his traces much earlier in [62]. The commutator subspace has been used previously to study spectral forms of the Dixmier traces, e.g. [6,89]. Outside of [6] the proof we have given of Connes’ original theorem is distinct from others, as it does not rely on the properties of the noncommutativeresidue (that it is the unique positive trace on classical pseudo-differentialoperators of order d )oron-function techniques. For the standard method of proving Connes’ original theorem, following [45], see the monograph [100, Theorem 7.18] or [4, §4.6]. We developed the result for non-measurable pseudo-differential operators in [126].

Spectral Formulas for the Noncommutative Residue The idea of the noncommutative residue as a spectral object appeared in [89, p. 359] and [6, Corollary 2.14], where the noncommutative residue was equated with a formula involving only the eigenvalues of a classical pseudo-differential operator of order d . The formula involving expectation values (Corollary 11.6.16) was obtained as a corollary in [126] of Nigel Section 11.8 Notes 381

Kalton’s results on modulated operators. The extension of the noncommutative residue to a vector-valued residue on any Hodge–Laplacian modulated operator by using the expectation values (Deﬁnition 11.6.6), and the subsequent ability to transfer this residue to (unital) noncommutative geometry, as discussed in the next chapter, is a new result.

Integration of Functions For the observation that Connes’ Trace Theorem can be used to recover the integral of a smooth function see, e.g. [100, Corollary 7.21], [9, §1.1], [143, p. 98], [189]. Connes used the - function residue to derive the integral on L1.X/ in [47]. Theorem 11.7.10 is a stronger variant of [151, Theorem 2.5]. In the cited paper we proved the statement of Theorem 11.7.10 for Dixmier traces associated to -function residues. The proof employing the methods of Hodge–Laplacianmodulated operators is different, and yields the result that integration of square integrable functions on closed manifolds is independent of the trace used. To investigatehow traces can recover integrationof functions that are not square d=2 integrable, the product Mf .1 g/ needs to be replaced by alternative symmetrized d=4 d=4 versions such as .1 g / Mf .1 g / . We refer the interested reader to [151]. Chapter 12 Residues and Integrals in Noncommutative Geometry

12.1 Introduction

In this chapter we deﬁne the noncommutative residue for an arbitrary unital spectral triple in the noncommutative, or quantum, calculus of Alain Connes, and we show how the noncommutative residue in noncommutative geometry is analogous to its classical counterpart. The quantum calculus is a direct analog of the pseudo-differential operator calculus of a closed Riemannian manifold. The spectral formula for the scalar noncommutative residue on a manifold obtained in Corollary 11.6.16 in Chapter 11 suggests the following construction. If H is a separable Hilbert space and D :Dom.D/ ! H is an unbounded self- adjoint operator with compact resolvent (such a pair .H , D/ is commonly called an unbounded Fredholm module), deﬁne 1 Xn 1 Res .A/ :D hAe , e i , A 2 M 1 D log.2 C n/ j j 1, j D0 nD0

1 where Œ denotes an equivalence class in l1=c0 and fej gj D0 is an orthonormal basis of H such that Dej D j ej for the eigenvalues j0jj1jj2j of D.We refer to the basis for H as in the last sentence, which is guaranteed by the fact that D has compact resolvent, as an ordered eigenbasis for the operator D. The ﬁrst task of this chapter is to show that the noncommutative residue in noncommutative geometry is analogous to its Riemannian geometry counterpart. An unbounded Fredholm module has dimension d>0ifhDid :D .1 C D2/d=2 belongs to the weak-l1 ideal of compact operators

L1,1 :DfA 2 L.H / : .A/ 2 l1,1g.

d=2 This is a direct analogy to the fact that .1g / 2 L1,1 for the Laplace–Beltrami operator g on a d-dimensional closed Riemannian manifold. If an unbounded Fred- holm module has dimension d then we show that ResD is a vector-valued linear form on all operators of order d in Connes’ quantum calculus, and that, if ! is an extended limit on l1 (or, equivalently, a state on the algebra l1=c0) then the formula

Tr!.A/ D !.ResD.A// holds for any Dixmier trace applied to an operator A of order d in the calculus. Section 12.1 Introduction 383

Example 10.3.7 in Chapter 10, and Section 11.7 in the last chapter, suggest that the functional

d d Int!.A/ :D Tr!.AhDi / D !.ResD.AhDi //, A 2 L.H / (12.1) should be viewed as the integral in Connes’ quantum calculus. We have to use distinct terminology for those A 2 L.H / such that the value Int!.A/ is independent of the choice of !.

Deﬁnition 12.1.1. Let .H , D/ be an unbounded Fredholm module of dimension d, d that is, hDi 2 L1,1.

(a) If A 2 L.H / we say the value Int!.A/ as above, where ! is an extended limit on l1, is the noncommutative integral of A.

(b) If the value Int!.A/ does not depend on the choice of extended limit !,thenwe say A is q-measurable and denote the value Int.A/.

Due to Lemma 9.7.4, the notion of q-measurability of the operator A coincides with the Dixmier measurability of the operator AhDid , where the notion of Dixmier measurability was defined in Definition 9.1.1. We investigate several aspects of the noncommutative integral, which, essentially, is the final investigation of the book. Assume that the spectrum of the operator D satisfies the following Weyl asymptotic condition, d jnj n, n 0.

This is not a standard imposition in noncommutative geometry, although common and frequently used examples of unbounded Fredholm modules satisfy this condition. The Weyl asymptotic condition is a very natural one given the intention that D2 be a noncommutative version of the Hodge–Laplacian g . If we use the following notation for the logarithmic means M : l1 ! l1, P n 1 xj Pj D0 j C1 .M.x//n :D n 1 , x 2 l1, n 0, j D0 j C1 then the noncommutative integral (12.1) has the following form in terms of convergence of logarithmic means of expectation values. Tauberian results can be used to obtain conditions when the noncommutative integral of a bounded operator is, in fact, the limit of its expectation values. The following theorem is the main result of this chapter. 384 Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.1.2. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl 1 asymptotic condition as above, and let fej gj D0 be an ordered eigenbasis for the operator D.

(a) For any extended limit ! on l1 the noncommutative integral Int! deﬁned in equation (12.1) is a singular state on the algebra L.H /, and the noncommutative integral of A 2 L.H / is given by an extended limit applied to the logarithmic mean of the expectation values of A,

1 Int!.A/ D .! ı M/.fhAen, enignD0/. (12.2)

(b) A bounded operator A 2 L.H / is q-measurable if and only if its expectation values logarithmically converge, and then

1 Int.A/ D .lim ıM/.fhAen, enignD0/.

lim .hAen, enihAem, emi/ D 0

when n>m!1such that log log n log log m ! 0,then

Int.A/ D lim hAen, eni. n!1

Another result we show in this chapter says that the noncommutative integral Int! is a hypertrace on a maximal C -algebra associated to trace class perturbations.

Deﬁnition 12.1.3. A state ' on L.H / is a hypertrace on a C -algebra N L.H / if '.AB/ D '.BA/ for all A 2 N and B 2 L.H /.

Theorem 12.1.4. Let .H , D/ be an unbounded Fredholm module of dimension d. Then d N :DfA 2 L.H / : ŒA, hDi 2 L1,1 \ .M1,1/0g

is the maximal C -subalgebra of L.H / such that Int! is a hypertrace on N for every dilation invariant extended limit ! on l1.Here.M1,1/0 is the ideal obtained by L closing the trace class operators 1 in the symmetric norm kkM1,1 . Theorem 12.1.2 and Theorem 12.1.4 are contained, and proven, in Section 12.3. We remark in that section on the notion of quantum ergodicity, which is the principle that the limit of the sequence of expectation values exists and provides a “measure” (in the sense of Theorem 12.1.2 (c)). We also remark in Section 12.3 on how Theorem 12.1.4 Section 12.2 The Noncommutative Residue in Noncommutative Geometry 385 implies the commonly known result that if ŒD, A is bounded for A belonging to a norm dense subset of a C -algebra N L.H /, then Int! is a hypertrace on N for all extended limits on l1. In Sections 12.4 and 12.5 we look at a few simple examples of the noncommutative residue in noncommutative geometry. We show that the noncommutative residue is an invariant of isospectral deformation. In the examples of Connes, Landi and Dubios- Violette, the noncommutative residue of the isospectral deformation of a pseudo-differential operator of order d can be calculated by the noncommutative residue of the pseudo-differential operator itself. Thus, for isospectral deformations, which presently are the main class of examples of noncommutative geometries, the Dixmier trace of operators in the quantum calculus and the quantum integral are calculated directly by symbols and the classical Lebesgue integral. We illustrate by considering the noncommutative torus, which is an isospectral deformation of smooth functions on the ordinary torus that is related to Weyl quantization. Finally, in Section 12.6, the last section of the book, we touch on some future applications of singular traces. The heat kernel formulation of Dixmier traces is associated to partition functions, or Gibbs states. The expectation value formulas mentioned above are associated to increasing to inﬁnity energy levels of quantum systems, or the notion of large quantum numbers. We reﬂect on how the coincidence of these formulations with singular traces, which has been derived in the different sections of this book, give a fundamental, and interesting, mathematical formalization of the link between high temperature limits as classical limits, the correspondence principle in quantum mechanics, and the notion of singular traces.

12.2 The Noncommutative Residue in Noncommutative Geometry

Chapter 11 introduced the vector-valued noncommutative residue of a Hodge–Lapla- cian modulated operator T : L2.X/ ! L2.X) on a closed Riemannian manifold X, 1 Xn 1 Res.T / D hTe , e i log.2 C n/ j j j D0 nD0

1 where Œ denotes an equivalence class in l1=c0 and fej gj D0 is an ordered eigenbasis of the Hodge–Laplacian g. When the Hodge–Laplacian modulated operator T is localized (see Deﬁnition 11.6.10), so that it locally has a modulated symbol, this spectral formula reduces to an asymptotic integral formula on the cotangent bundle. In particular, Wodzicki’s noncommutative residue on classical pseudo-differential operators of order d becomes a special case of the spectral formula. This opens the ﬁeld for an analogous notion of noncommutative residue in spectral extensions of differential geometry, such as Alain Connes’ noncommutative geometry. 386 Chapter 12 Residues and Integrals in Noncommutative Geometry

Connes’ calculus is based on spectral triples, .A, H , D/.HereA is an -algebra of bounded operators on the Hilbert space H . Unlike the commutative algebra of functions C 1.X/ on a closed Riemannian manifold X, where functions act by pointwise multiplication on the Hilbert space L2.X/, the algebra A can be noncommutative. The operator D :Dom.D/ ! H is self-adjoint with compact resolvent. It is intended to be the analog of a ﬁrst order Dirac operator (mainly inspired by the Hodge–Dirac operator or the Dirac operator of a spin manifold [146]), and the positive operator D2 is the analog of the Hodge– Laplacian. The “analogy” comes in the form of replicating the algebraic and spectral properties of the operators of differential geometry, as below. Predominately we are interested in the data .H , D/, which, when singled out, is called an unbounded Fredholm module. Given an unbounded Fredholm module .H , D/, the functional calculus of self- adjoint operators allows us to deﬁne the positive operator with trivial kernel,

hDis :D .1 C D2/s=2, s 2 R.

When s<0 this operator (extends to) a compact operator on H . Sobolev spaces can be generalized by the Connes–Moscovici approach. Set

H s :D Dom.hDis/, s>0, with norm s s kxkH s :DkhDi xk, x 2 H .

Let op m, m 2 R, denote the set of continuous linear operators A : H s ! H sm, 8s 2 R.Thenop m contains the noncommutative versions of pseudo-differential operators [54], (compare also with Proposition 10.3.4). An unboundedFredholm module .H , D/ is said to have dimension d>0if

d hDi 2 L1,1.

Henceforth we assume that all unbounded Fredholm modules have dimension d. The analogy of the graded algebra formed by op m, m 2 R, to the pseudo-differential operator calculus is given in the next lemma.

Lemma 12.2.1. Let .H , D/ be an unbounded Fredholm module of dimension d as above. (a) If A 2 op 0 then A is a bounded operator on H . Section 12.2 The Noncommutative Residue in Noncommutative Geometry 387

(b) If A 2 op m, m<0,thenA is a compact operator, and if A 2 op m, m<d, then A is a trace class operator.

d (c) If A 2 op then A 2 L1,1.

Proof. Suppose A 2 op 0.SinceH 0 H then A : H ! H is continuous and linear. This proves (a). If A belongs to op m, m 2 R, then note that AhDim and hDimA belong to op 0. In which case there exists B 2 op 0 such that A D BhDim.SincehDim is compact, m d d m<0, it follows that A 2 op is compact. Since hDi 2 L1,1 then A 2 op belongs to L1,1. This proves (c). If m<d then hDim D .hDid /r ,wherer D m.d/1 > 1. By Lemma 8.6.2, m m hDi 2 L1 and it follows that A 2 op , m<d, is trace class.

There is a wider class of operators than those in op d that we will deal with simultaneously. The following operators are the analog of the Hodge–Laplacian modulated operators introduced in Chapter 11.

Deﬁnition 12.2.2. Let .H , D/ be an unbounded Fredholm module of dimension d. Then A 2 L.H / is .H , D/-modulated (equivalently, hDid -modulated in the sense of Deﬁnition 11.2.1)if

1=2 d 1 kAkmod :D sup t kA.1 C thDi / k2 < 1. t>0

The definition, while useful in an abstract sense, is not always convenient to test. Section 11.3 noted that there was a more convenient formula, involving symbols, by which to define a Laplacian modulated operator. In noncommutative geometry there are no symbols, the replacements are spectral formulas. An orthonormal basis 1 of fej gj D0 is called an ordered eigenbasis of D if Dej D j ej , j 0, where j0jj1j are the eigenvalues of D order by increasing absolute value (note this is possible since D has compact resolvent). The eigenvalues of D will always appear ordered by increasing absolute value. When a Weyl asymptotic condition is satisfied by the eigenvalues (as in the statement below) the modulated condition is more conveniently tested by the action of a bounded operator A on the ordered eigen- 1 basis fej gj D0. Compare the next lemma with Lemma 11.3.13.

Lemma 12.2.3. Let .H , D/ be an unbounded Fredholm module of dimension d sat- d 1 isfying the Weyl condition (i.e. hni n, n 0), and let fej gj D0 be an ordered eigenbasis of D as above. Then A 2 L.H / is .H , D/-modulated if and only if

X1 2 1 kAej k D O..1 C n/ /, n 0. j DnC1 388 Chapter 12 Residues and Integrals in Noncommutative Geometry

Proof. Fix n 0andletPn be the projection onto the linear span of vectors ej , j n C 1. Then X1 2 2 kAPnk2 D kAej k . j DnC1

d Since hDi 2 L1,1 satisﬁes the Weyl condition, it follows that there exist constants c2 >c1 > 0 such that c1 c2 E d 0, P E d 0, . hDi n C 1 n hDi n C 1

The assertion now follows from Lemma 11.2.5.

We summarize the properties that can be found in Section 11.2 concerning modu- d lated operators, noting that, by assumption, 0 hDi 2 L1,1.

Lemma 12.2.4. Let .H , D/ be an unbounded Fredholm module of dimension d, and 1 let fej gj D0 be an ordered eigenbasis of D as above. L 1 L (a) If A 2 .H / is .H , D/-modulated, then A,diag.fhAen, enignD0/ 2 1,1. (b) If A 2 L.H / is .H , D/-modulated, then

Xn hAej , ej iDO.log.2 C n//, n 0. j D0

(d) If A 2 op d then A is .H , D/-modulated.

Proof. The assertion of (a) is directly shown by Theorem 11.2.3. The assertion of (b) follows by arguing as in the proof of Lemma 11.6.5. The assertion of (c) follows from Proposition 11.2.2. The assertion of (d) follows from (c) since AhDid 2 L.H / if A 2 op d .

Part (b) of the lemma shows that we can deﬁne a vector-valued noncommutative residue which is a linear form on operators that are .H , D/-modulated. Part (d) says that the residue is a linear form on operators of order d in the quantum calculus.

Deﬁnition 12.2.5. Let .H , D/ be an unbounded Fredholm module of dimension d, 1 and let fej gj D0 be an ordered eigenbasis of D as above. Deﬁne the noncommutative Section 12.2 The Noncommutative Residue in Noncommutative Geometry 389 residue associated to .H , D/ by 1 Xn 1 Res .A/ :D hAe , e i , A 2 L 1, D log.2 C n/ j j 1, j D0 nD0 where Œ denotes an equivalence class in l1=c0.

If the equivalence class ResD.A/ contains a constant sequence, then we say that ResD.A/ is scalar valued. The bijection between scalars and equivalence classes of constant sequences will be implicit. The main result on modulated operators (Theorem 11.2.3) is central to connecting the noncommutative residue associated to an unbounded Fredholm module to the established use of the Dixmier trace in noncommutative geometry (NCG). The connection places the noncommutative residue at the heart of the integration half of Connes’ quantized calculus. Since the proof of the next theorem is identical to that of Corol- lary 11.2.4 we omit it. Recall from Lemma 9.7.4 that every extended limit ! on l1 can be used to deﬁne a Dixmier trace on the ideal L1,1. Recall also that a trace on L 1 1 1,1 is normalized if it takes the value 1 on the operator diag.f.n C 1/ gnD0/. Theorem 12.2.6 (Trace Theorem NCG). Let .H , D/ be an unbounded Fredholm module of dimension d. Then every .H , D/-modulated operator A belongs to L1,1 and

(a) for a Dixmier trace Tr!,where! is an extended limit on l1,

Tr!.A/ D !.ResD.A//.

(b) A is Dixmier measurable (takes the same value for every Dixmier trace) if and only if ResD.A/ is scalar valued (and then

Tr!.A/ D ResD.A/,

for every extended limit ! on l1 ). (c)

'.A/ D ResD .A/

for a scalar value ResD.A/ for all normalized traces ' : L1,1 ! C if and only if Xn hAej , ej iDResD.A/ log.n/ C O.1/ , n 1, j D0

1 for a scalar value ResD.A/,wherefengnD0 is an ordered eigenbasis of D. 390 Chapter 12 Residues and Integrals in Noncommutative Geometry

Part (a) of the theorem indicates that Dixmier traces are the only traces whose value on .H , D/-modulated operators can be obtained by combining the noncommutative residue with an extended limit. We will not discuss further in this chapter the stronger version of measurability given in Part (c) of the theorem. It could be argued, given the results in Chapter 11, that considering this stronger version of measurability is warranted. However, we will use the “traditional” form of the scalar noncommutative residue based on Dixmier traces. We restrict ourselves to the comment that one could use the set of all normalized traces on L1,1 instead of the set of Dixmier traces and make adjusted statements in what follows.

Remark 12.2.7. The convenience of the noncommutative residue is that it is a unique class. The disadvantage is that it is vector-valued. The reduction of the vector-value to a unique scalar value is not always possible (not all .H , D/-modulated operators are measurable). Connes’ notion of measurability [48], as studied in Chapter 9 of this book, is revealed as precisely the condition for ResD to be scalar valued.

12.3 The Integral in Noncommutative Geometry

1 1 If g : C .X/ ! C .X/ is the Laplace–Beltrami operator on a closed d-dimensional Reimannian manifold X, then Example 10.3.7 in Chapter 10, and Section 11.7 in the last chapter, showed that the functional

d=2 d=2 A ! Tr!.A.1 g/ / D !.Res.A.1 g/ // provides, when A D Mf , f 2 L2.X/, integration of square integrable functions on X. It should not be overlooked that the same functional, when A is a pseudo- differential operator of order 0, is a trace on the -algebra of pseudo-differential operators of order 0. Thus, “commutative” geometry, which is sometimes how differential geometry is referred to within noncommutative geometry, contains the precedent of the above functional providing a trace on a noncommutative algebra. If .H , D/ is an arbitrary unbounded Fredholm module of dimension d, the observations above suggest that the functional

d d Int!.A/ :D Tr!.AhDi / D !.ResD .AhDi //, A 2 L.H /, (12.3) be viewed as the integral in Connes’ quantum calculus. For convenient normalization we assume that Int!.1/ D 1, or, what is the same condition (see Theorem 9.7.5), Xn d d 1 d Tr!.hDi / D ResD.hDi / D lim hj i D 1, n!1 log.2 C n/ j D0 for every extended limit ! on l1. In this section we examine the spectral properties of the integral in (12.3), and its association with the logarithmic means of expecta- Section 12.3 The Integral in Noncommutative Geometry 391 tion values of the operator A 2 L.H /. This “quantum ergodic” property is a spectral equivalent of the Lebesgue integral. We will also show that Int!, in analogy to its differential geometry prototype, is a trace on the algebra of operators of order 0 in the Connes–Moscovici quantum calculus. We recall from Deﬁnition 12.1.1 that q-measurable operators are those operators A 2 L.H / such that the value Int!.A/ is independent of the choice of !.Inthatcase we write Int.A/ for the unique value. Since essentially bounded functions and classical zero order pseudo-differential operators are examples of q-measurable operators in differential geometry, it is natural in noncommutative geometry to “impose” the condition of q-measurability on the algebra A in a spectral triple .A, H , D/. The next theorem characterizes what is being imposed. Assuming Weyl asymptotics, it shows that q-measurability of a bounded operator is equivalent to logarithmic convergence of its expectation values.

Theorem 12.3.1. Let .H , D/ be an unbounded Fredholm module of dimension d. d Suppose that the operator D satisﬁes the Weyl asymptotic condition, hni n, n 1 0.LetfengnD0 be an eigenbasis of D such that Den D nen where j0jj1j are increasing.

(a) If A 2 L.H /, then, for every extended limit ! on l1, we have

1 Int!.A/ D .! ı M/.fhAen, enignD0/.

L 1 (b) An operator A 2 .H / is q-measurable if and only if the sequence fhAen, enignD0 of expectation values logarithmically converge, and then

1 Int.A/ D .lim ıM/.fhAen, enignD0/.

Proof. Let A 2 L.H /. By Lemma 12.2.4(c) the operator AhDid is .H , D/-modulated, so Theorem 12.2.6(a) provides the formula 1 Xn Int .A/ D ! hAhDid e , e i ! log.2 C n/ j j j D0 1 Xn D ! h id hAe , e i . log.2 C n/ j j j j D0

d By the Weyl asymptotic condition, we have hj i D .1 C o.1//=.j C 1/, j 0. Hence, Xn Xn 1 h id hAe , e iD hAe , e iCo.log.n//. j j j j C 1 j j j D0 j D0 392 Chapter 12 Residues and Integrals in Noncommutative Geometry

Since ! is an extended limit, then P Xn n hAej ,ej i 1 hAe , e i j D0 C Int .A/ D ! j j D ! P 1 j . ! log.2 C n/ 1 C j n 1 j D0 j D0 1Cj

The last equality follows by applying Lemma 6.3.5, since P n 1 j D0 C lim 1 j ! 1C. n!1 log.2 C n/

The assertion of (a) is shown. The assertion of (b) follows from (a) and Lemma 9.3.6.

The observations in Theorem 12.3.1 bring the noncommutative integral into the realm of the classical theory of divergent sequences. Tauberian theorems for the logarithmic mean can be used to determine if the sequence of expectation values converges. The ordinary limit calculates the noncommutative integral in this case. We give an example of a Tauberian condition, the condition of slow oscillation at inﬁnity [142], and also [163].

Corollary 12.3.2. Let .H , D/ be an unbounded Fredholm module of dimension d 1 satisfying the Weyl asymptotic condition, and let fengnD0 be an ordered eigenbasis of D.IfA is q-measurable and the expectation values of A 2 L.H / have slow oscillation at inﬁnity, meaning that

lim .hAen, enihAem, emi/ D 0 when n>m!1such that log log n log log m ! 0,then

Int.A/ D lim hAen, eni. n!1

1 Proof. Let sn :DhAen, eni and s DfsngnD0.IfA is q-measurable then s is logarithmically convergent. Using Kwee’s Tauberian result [142, Lemma 3], and the remark [163, Remark 3], the condition of slow oscillation at inﬁnity of s implies that s is convergent. The logarithmic means are regular, so if s is convergent then .lim ıM/.s/ D lim s and the assertion is shown.

Despite us imposing the condition that the operator D satisﬁes the Weyl asymptotic d d condition, hni n, n 0, which implies that hDi is Dixmier measurable in the sense of Chapter 9, we note that there are many operators A 2 L.H / which are not q-measurable. The expectation

1 L A !fhAen, enignD0 , .H / ! l1, Section 12.3 The Integral in Noncommutative Geometry 393 is onto, so it sufﬁces, to know that there are bounded operators that are not q-measurable, to know only that there are bounded sequences that are not logarithmically convergent.

Example 12.3.3. Suppose that the Weyl asymptotic condition is satisﬁed and A 2 L.H / is such that

X1 hAe , e iD 2j 2j C1 .n/, n 0. n n Œ22 ,22 / j D0

Then A is not q-measurable. P 1 1 Proof. 2j 2j C1 Set xn :D j D0 Œ22 ,22 /.n/, n 0, and x DfxngnD0.Sincekxk1 1, then x 2 l1. We show that M.x/ has convergent subsequences which converge to different values, hence M.x/ 62 c. Consider the subsequence

2n 2mC1 X22 Xn1 22X 2n 1 x 1 1 M.x/.22 / D k C o.1/ D C o.1/ 22n 22n log.2 / k C 1 log.2 / m k C 1 kD0 mD0 kD222 Xn1 Xn1 1 22m 1 2m D n log.2 / C o.1/ D 2 log.2/ C o.1/ log.222 / 22n log.2/ mD0 mD0 1 D C o.1/, n 0. 3 Similarly,

2nC1 2 M.x/.22 / D C o.1/, n 0. 3 It follows that x is not logarithmically convergent.

We now consider under what conditions the noncommutative integral Int! is a hypertrace on a C -subalgebra N L.H /. Hypertraces were deﬁned in Deﬁni- tion 12.1.3.

Theorem 12.3.4. Let .H , D/ be an unbounded Fredholm module of dimension d (a Weyl asymptotic condition on D is not assumed). Then

d N :DfA 2 L.H / : ŒA, hDi 2 L1,1 \ .M1,1/0g

is the maximal C -subalgebra of L.H / such that Int! is a hypertrace on N for every extended limit ! on l1.Here.M1,1/0 is the ideal obtained by closing the trace class operators in the symmetric norm kkM1,1 . 394 Chapter 12 Residues and Integrals in Noncommutative Geometry

Proof. First, we show that N is indeed a C -algebra. We omit showing that N is a -algebra, as these properties are an easy calculation using the fact that L1,1 \ .M1,1/0 is a two-sided ideal. If T 2 L1,1, recall the deﬁnition of the quasi-norm 1 kT k1,w :D supn0.1 C n/ .n, T/. Evidently, kATBk1,w kAk1kT k1,wkBk1 for all A, B 2 L.H /. Let An ! A as n !1in the uniform topology where An 2 N .ThenA 2 L.H / satisﬁes

d d d kŒhDi , A ŒhDi , Ank1,w 4kA Ank1khDi k1,w ! 0.

d d By assumption ŒhDi , An 2 L1,1 \ .M1,1/0, n 0, hence ŒhDi , A 2 L1,1 \ .M1,1/0 since L1,1 \ .M1,1/0 is closed in the quasi-norm topology. It follows that A 2 N . Observe that

d d d d Int!.ŒA, B/ D Tr!.ABhDi BAhDi / D Tr!.BhDi A BAhDi / d D Tr!.BŒhDi , A/, A, B 2 L.H / (12.4) by using the trace property of a Dixmier trace Tr!. d Now suppose that ŒhDi , A 2 L1,1 \ .M1,1/0.SinceL1,1 \ .M1,1/0 is an ideal, it follows that

d BŒhDi , A 2 L1,1 \ .M1,1/0, B 2 L.H /.

d Hence, we have that Tr!.BŒhDi , A/ D 0 since all Dixmier traces, as continuous traces on M1,1,vanishonL1,1 \ .M1,1/0. From (12.4), it follows that Int! is a hypertrace on N for every extended limit !. Conversely, if Int! is a hypertrace on N for every extended limit ! then, from (12.4),

d Tr!.BŒhDi , A/ D 0 for all Dixmier traces and every B 2 L.H /. In particular, choose B such that BŒhDid , A DjŒhDid , Aj.So

d Tr!.jŒhDi , Aj/ D 0 for all Dixmier traces. We recall from Theorem 10.1.3 (or Theorem 9.2.1) that this d condition implies that ŒhDi , A 2 L1,1 \ .M1,1/0. Corollary 12.3.5. Let .H , D/ be an unbounded Fredholm module of dimension d. For any extended limit ! on l1, (a) Int! is a hypertrace on the C -algebra

N :D fA 2 L.H / : ŒD, A 2 L.H /g. Section 12.3 The Integral in Noncommutative Geometry 395

(b) Int! is a hypertrace on the C -algebra

N :D fA 2 L.H / : ŒhDi, A 2 L.H /g.

Proof. Let d D kr, k 2 N and k>d.Then0

ŒhDid , A DhDid ŒhDid , AhDid Xk DhDid hDir.ki/ŒhDir , AhDir.i1/ hDid iD1 Xk D hDiirŒhDir , AhDi.kC1i/r. iD1

Suppose that ŒD, A is bounded. It follows (see [42, 248] or [186, Theorem 2.4.3]) that ŒhDir , A is bounded. The individual terms in the sum satisfy, for 1 i k,

ir r .kC1i/r hDi ŒhDi , AhDi 2 L d L d . ir ,1 .kC1i/r ,1

If B1 2 L d and B2 2 L d ,then ir ,1 .kC1i/r ,1

ir=d .ik1/r=d .n, B1/ const .n C 1/ , .n, B2/ const .n C 1/ , n 0.

Hence, Z Z 1 1 .kC1/r=d kB1B2k1 .s, B1/.s, B2/ds const .s C 1/ ds < 1. 0 0

d Hence ŒhDi , A is trace class. By Theorem 12.3.4 the state Int! is a hypertrace for the -algebra of operators A 2 L.H / such that ŒD, A is bounded. It follows from Theorem 12.3.4that Int! is a hypertrace for the operator norm closure of this -algebra. This proves the assertion of (a). The assertion of (b) follows from (a) (as applied to the operator hDi instead of D).

Remark 12.3.6. Alain Connes and Henri Moscovici [54], introduced zero order noncommutative pseudo-differential operators as those A 2 op 0 such that the operators k k th ıhDi.A/, k 0, are bounded where ıhDi denotes the k power of the derivation ıhDi.A/ :D ŒhDi, A. We can then observe that, by Corollary 12.3.5 (b), the noncommutative integral Int! is a hypertrace on the zero order operators in the Connes– Moscovici pseudo-differential calculus. 396 Chapter 12 Residues and Integrals in Noncommutative Geometry

12.4 Example of Isospectral Deformations

The most commonly studied noncommutative geometries lie within the category of isospectral deformations of commutative geometries. This section shows that the noncommutative residue is an invariant of isospectral deformation. Let X be a closed d-dimensional Riemannian manifold where d 2, with associated unbounded Fredholm module p .L2.X/, g /

1 1 where g : C .X/ ! C .X/ is the Laplace–Beltrami operator associated to a ﬁxed metric g (see Section 10.3 and Section 11.6). We scale the Laplace–Beltrami operator, as given in Deﬁnition 10.3.3, by the factor Vol.X/ Vol.Sd1/ 2=d 1 Vol.X/ 2=d D d.2/d 4 .d=2 C 1/

d=2 so that Tr!..1 g / / D 1. Usually, the Fredholm module would be formed from the pair of the Hodge–Dirac operator and the Hilbert space of square integrable sections of the exterior bundle. However, to considerp the noncommutative residue it is sufﬁcient to consider the moduleabove. The .L2.X/, g /-modulated operators, in the terminology of Section 12.2, are the Hodge–Laplacian modulated operatorsp introduced in Section 11.6, and the noncommutativeresidue associated to .L2.X/, g / is the residue on Hodge–Laplacian operators introduced in Deﬁnition 11.6.6. Let Iso.X/ denote the compact Lie group of isometries of the closed manifold X [133, 167]. We recall that an isometry ˛ : X ! X is a diffeomorphism which preserves the metric g. The topology on Iso.X/ is the compact-open topology. The group Iso.X/ acts on the Hilbert space L2.X/ by the unitaries

1 .V˛f /.x/ :D f.˛ .x//, f 2 L2.X/, ˛ 2 Iso.X/.

The unitary V˛ commutes with the Laplacian for any ˛ 2 Iso.X/,

Œg , V˛ D 0.

This commutation property is the deﬁning feature of an isometry amongst other dif- feomorphisms [111, 250]. Suppose that we have a smooth action by isometries of the r-torus Tr , r 2, on the manifold X (equivalently a smooth embedding of Tr in r Iso.X/ such that the map .s, x/ ! ˛s.x/, s 2 T , x 2 X, is smooth in s and x). Denote r .Vsf /.x/ :D f.˛s .x//, f 2 L2.X/, s 2 T . Section 12.4 Example of Isospectral Deformations 397

Using this action, we define an isospectral deformation. The deformation involves the discrete version of the well-known Moyal product [95, 192]. Define, Z 2ihm,si r Um :D e Vsds, m 2 Z (12.5) Tr where the integral converges in the strong sense since Vs is a strongly continuous r unitary representation of T on L2.X/ (shown below in Lemma 12.4.5). We also show r below that the operators Um, m 2 Z , form a complete system of pairwise orthogonal projections. Let ƒ denote the additive group of skew symmetric matrices in Mr .T/, and suppose that ‚ 2 ƒ.IfA 2 L.L2.X//, then define the isospectral deformation of A,formally,by X ihn,‚.m/i L‚.A/ :D e UmAUn. (12.6) m,n2Zr

To associate this formal deﬁnition to a linear operator L‚.A/, the convergence of the square partial sums X ihn,‚.m/i e UmAUn, k 0, (12.7) m,n2Œk,kr needs to be considered. The partial sums are bounded operators, but they may not converge in the uniform or strong sense when ‚ 6D 0. The following theorems are the main results of this section. They state that the noncommutative residue is an invariant of isospectral deformation, and, further, that the isospectral deformation of Hodge– Laplacian modulated operators is a complete trace invariant.

Theorem 12.4.1. If A 2 L.L2.X// is a Hodge–Laplacian modulated operator, then L‚.A/ 2 L.L2.X// is a Hodge–Laplacian modulated operator (where L‚.A/ is the L2-limit of the partial sums in (12.7) in the ideal of Hilbert–Schmidt operators), and

Res.L‚.A// D Res.A/, 8‚ 2 ƒ.

In particular, if A : C 1.X/ ! C 1.X/ is a classical pseudo-differential operator of order d,then

Res.L‚.A// D ResW .A/, 8‚ 2 ƒ where ResW is Wodzicki’s noncommutative residue on the closed Riemannian manifold X. The proof of the theorem is given below. First we show that a stronger statement about trace invariance is true. If g is the Laplace–Beltrami operator, we recall from d=2 Section 11.6 that 0 .1g/ 2 L1,1 and that any Hodge–Laplacian modulated operator belongs to the ideal L1,1. 398 Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.4.2. If A 2 L.L2.X// is a Hodge–Laplacian modulated operator, then A, L‚.A/ 2 L1,1 and

'.L‚.A// D '.A/, 8‚ 2 ƒ for every trace ' : L1,1 ! C. In particular, Tr!.L‚.A// D Tr!.A/ 8‚ 2 ƒ, for every Dixmier trace Tr!, and L‚.A/ is Dixmier measurable if and only if A is Dixmier measurable. This result allows us to calculate the noncommutative integral of the deformation 1 L‚.Mf /, Mf 2 L.L2.X//, of a smooth function f 2 C .X/. Note that we shorten 1 L‚.Mf / to L‚.f /.ThenL0.f / D Mf , f 2 C .X/. Fix the matrix ‚ 2 ƒ.The following characterization of the isospectral deformation of a smooth function is the Peter–Weyl form [53].

1 Theorem 12.4.3. If f 2 C .X/ then L‚.f / 2 L.L2.X// where L‚.f / is the strong limit of the partial sums in (12.7). Equivalently, (a) X

L‚.f / D MUmf V 1 , 2 ‚.m/ m2Zr

where L‚.f / is the uniform limit of the partial sums of the right-hand side of the display.

(b) The operator L‚.f / is a q-measurable operator with Z 1 Int.L‚.f // D f.u/du. Vol.X/ X

We observe, as a corollary to Theorem 12.4.2, that L‚.A/ can be associated to a closeable unbounded operator for any bounded operator A 2 L.L2.X//,andthat the formula (12.3) can be used to extend the noncommutative integral to unbounded operators of the form L‚.A/ .

Corollary 12.4.4. If A 2 L.L2.X//,thenL‚.A/ deﬁnes a continuouslinear operator

d L‚.A/ : H .X/ ! L2.X/,

d where H .X/ denotes the Sobolev space of order d described in Section 10.3 (L‚.A/ d is the strong limit of the square partial sums on H .X/). The operator L‚.A/.1 d=2 g/ has a bounded extension as a Hodge–Laplacian modulated operator, and deﬁning Int! on L‚.A/ by the same formula as in (12.3), we have

Int!.L‚.A// D Int!.A/. Section 12.4 Example of Isospectral Deformations 399

We now prove the results.

r Lemma 12.4.5. The action s ! Vs, s 2 T , is strongly continuous. Proof. It is sufﬁcient to show continuity as s ! 0, and on a connected component of X. Suppose Xi is a connected component. We claim that k.Vs 1/f k2 ! 0ass ! 0 1 for every function f 2 C .Xi /.Wehave

k.Vs 1/f k2 Vol.Xi / k.Vs 1/f k1 kf kLip sup dist.x, ˛s.x//. x2Xi

Then k.Vs 1/f k2 ! 0ass ! 0since˛s is smooth. The claim is shown. 1 Now let fn 2 C .Xi / converge to f 2 L2.Xi / as n !1in the L2-norm. Then

lim k.Vs 1/f k2 lim k.Vs 1/fnk2 C 2kfn f k2 D 2kfn f k2. s!0 s!0

Since n can be arbitrarily large, it follows that s ! Vs is strongly continuous on L2.Xi /. N Since X D[iD1Xi is a ﬁnite disjoint union of connected components, then N L2.X/ D˚iD1L2.Xi / and it follows that s ! Vs is strongly continuous on L2.X/.

The preceding lemma justiﬁes the deﬁnition of the operators Um given in (12.5).

Zr Lemma 12.4.6.P The operators Um, m 2 , are pairwise orthogonal projections and the series m2Zr Um converges to 1 strongly. Proof. It follows from (12.5) that, for m, n 2 Zr , Z Z 2i.hn,tiChm,si/ 2i.hn,tiChm,si/ UmUn D e VsVt dsdt D e VsCt dsdt. T2r T2r

Using the substitution t ! t s, Z 2i.hn,tiChmn,si/ UmUn D e Vt dsdt T2r Z Z 2ihmn,si 2ihn,ti D e ds e Vt dt D ı0.m n/Un. Tr Tr

Similarly, we have Z Z 2ihm,si 2ihm,ui Un D e Vsds D e Vudu D Un. Tr Tr

r Hence, the operators Um, m 2 Z , are pairwise orthogonal projections. 400 Chapter 12 Residues and Integrals in Noncommutative Geometry

P We now prove that the series m2Zr Um converges to 1 strongly. The Hilbert space L2.X/ can be decomposed into a direct sum of eigenspaces for g . Since the unitaries r r Vs, s 2 T , commute with g, each of these eigenspaces is Vs-invariant, s 2 T .Since each of these eigenspaces is ﬁnite dimensional, there exists a common eigenbasis en, r n 0, for Vs, s 2 T . For every n 0, the r-torus acts on the 1-dimensional space spanned by en according to the formula

2ihm,si r Vsen D e en, s 2 T ,

r for some m 2 Z (m depends on n). It is now clear from (12.5) that Umen D en.Since en, n 0, form an orthonormal basis of L2.X/, the assertion follows.

LemmaP 12.4.7. Let pk, k 0, be pairwise orthogonal projections such that 1 kD0 pk D 1. If j˛k,l jD1 for k, l 0, then the mapping X1 A ! ˛k,l pkApl , A 2 L2 k,lD0 is an isometry of L2 into itself. Here, the series converges in L2.

Proof. Fix a common eigenbasis en, n 0, for the projections pk, k 0. Choose sets Ak such that pken D en for n 2 Ak and pnek D 0otherwise.ForA 2 L2 and anm :DhAen, emi,wehavethat X1 X1 X X1 2 2 2 2 kAk2 D janmj D janmj D kpkApl k2. n,mD0 k,lD0 n2Ak,m2Al k,lD0 L 1 In other words, fpk 2pl gk,lD0 is an orthogonal decomposition of the Hilbert space L2. The assertion follows.

We now show that, in the case where A is Hilbert–Schmidt, the series in the formal deﬁnition (12.6) converges to a Hilbert–Schmidt operator. Fix a matrix ‚ 2 ƒ.

Corollary 12.4.8. For every A 2 L2, we have L‚.A/ 2 L2 and kAk2 DkL‚.A/k2. The series (12.6) converges in L2.

Proof. The proof follows from a combination of (12.5), Lemma 12.4.7 and Lem- ma 12.4.6.

Using the last lemma we can show, in the case where A is a Hodge–Laplacian modulated operator, that the square partial sums in the formal deﬁnition (12.6) converge in L2 to a Hodge–Laplacian modulated operator. Section 12.4 Example of Isospectral Deformations 401

Lemma 12.4.9. If A 2 L2 is Hodge–Laplacian modulated, then L‚.A/ is Hodge– Laplacian modulated and kL‚.A/kmod DkAkmod, where L‚.A/ is the Hilbert– Schmidt operator from Corollary 12.4.8.

Proof.LetA be Hodge-Laplacian modulated. From Section 11.2 we know that A is Hilbert–Schmidt and so, by Corollary 12.4.8, L‚.A/ is Hilbert–Schmidt. For brevity, d=2 1 denote Wt :D .1 t.1 g/ / 2 L.L2.X//, t>0. Note that ŒWt , Vs D 0, r r s 2 T , t>0, since Œg , Vs D 0, s 2 T . Hence, taking into account that Wt also 2 commutes with Um, m 2 Z ,wehavethat

L‚.A/Wt D L‚.AWt / 2 L2. By Corollary 12.4.8 again

kL‚.A/Wt k2 DkL‚.AWt /k2 DkAWt k2 Hence, 1=2 1=2 kL‚.A/kmod D sup t kL‚.A/Wt k2 D sup t kAWt k2 DkAkmod. t>0 t>0 We prove the main results.

Proof of Theorem 12.4.1. By assumption the operator A 2 L.L2.X// is Hodge– Laplacian modulated, and then Lemma 12.4.9 proves that L‚.A/ is Hodge–Laplacian modulated where L‚.A/ is the Hilbert–Schmidt operator from Corollary 12.4.8. The ﬁrst assertion of Theorem 12.4.1 follows. r r Recall that the unitaries Vs, s 2 T , and hence the projections Um, m 2 Z ,com- r mute with g . Select a common eigenbasis en, n 0, for g and Um, m 2 Z ,where 2 gen D nen and n, n 0, are increasing. Let pN , N 0, be the projection onto the linear span of fe0, :::, eN g.SinceXpN is a ﬁnite projection, we have ihn,‚.m/i Tr.L‚.A/pN / D e Tr.UmAUnpN /. m,n2Zr

By construction, pN commutes with the (pairwise orthogonal projections) Um, m 2 Zr . Hence, Tr.U AU p / D 0form ¤ n. Therefore, m n N X Tr.L‚.A/pN / D Tr.AUnpN / D Tr.ApN / 2Zr P n since n2Zr Un converges strongly to 1 and Tr is strongly continuous. This proves that 1 1 Res.L‚.A// D Tr.L‚.A/pN / log.2 C Tr.pN // N D0 1 1 D Tr.ApN / D Res.A/. log.2 C Tr.pN // N D0 402 Chapter 12 Residues and Integrals in Noncommutative Geometry

The assertions in Theorem 12.4.1 are proved. The particular statement when A is a classical pseudo-differential operator of order d nowfollows from Corollary 11.6.15.

Proof of Theorem 12.4.2. By assumption the operator A is V -modulated where V :D d=2 .1g/ . It follows from Lemma 12.4.9 that the operator L‚.A/ is V -modulated. By Theorem 11.2.3, we have that A and L‚.A/ belong to L1,1 and, by the choice of basis, that XN XN .k, L‚.A// D Tr.L‚.A/pN / C O.1/, .k, A/ D Tr.ApN / C O.1/, kD0 kD0 where pN is the projection in the proof of Theorem 12.4.1 above. Recall from the above proof that Tr.L‚.A/pN / D Tr.ApN /. Now, by Lemma 5.7.5, we have that XN .k, L‚.A/ A/ D O.1/. kD0

It follows from Theorem 5.7.6 that L‚.A/ A 2 Com.L1,1/. All the assertions in the theorem now follow.

We now prove Theorem 12.4.3.

Lemma 12.4.10. For f 2 C 1.X/, we have that X kUmf k1 < 1. m2Zr

r Proof. Deﬁne the function h : T X ! C by setting h.s, x/ :D f.˛s .x//.By the assumptions on f and the assumption of a smooth action of the torus on X, h is Tr Zr a smooth function on X. Select 0 ¤ m D .m1, :::, mr / 2 and let jmkj be the maximal term amongst jm1j, :::, jmr j. Using (12.5) together with integration by parts, we obtain Z 2ihm,si .Umf /.x/ D e h.s, x/ds r T Z rC1 r1 2ihm,si @ D .2imk/ e C h.s, x/ds. Tr r 1 @sk Therefore, for some constant depending on the function f ,wehave r1 r1 kUmf k1 .2jmkj/ khkC rC1.Tr X/ const hmi . The statement follows since hmir1, m 2 Zr , is summable. Section 12.4 Example of Isospectral Deformations 403

Lemma 12.4.11. For every f 2 L1.X/ and g 2 L2.X/, we have

r .UmMf Un/g D .Ung/.Umnf/, m, n 2 Z .

r Proof. For every g 2 L2.X/, we have from (12.5) that, for m, n 2 Z , Z 2i.hm,siChn,ti/ .UmMf Un/g D e VsMf Vt gdsdt. T2r

However, for s, t 2 Tr ,

VsMf Vt g D .VsMf /.g ı ˛t / D Vs.f .g ı ˛t // D .Vsf /.Vs.g ı ˛t // D .Vsf /.VtCsg/ and, therefore, Z 2i.hmn,siChn,tCsi/ .UmMf Un/g D e .Vsf /.VtCsg/dsdt ZT2r 2i.hmn,siChn,ui/ D e .Vsf /.Vug/dsdu TZ2r Z 2ihmn,si 2ihn,ui D e Vsfds e Vugdu Tr Tr

D .Umnf /.Ung/.

Proof of Theorem 12.4.3. Fix k, l 0and‚ 2 ƒ. It follows from (12.5) and a simple computation that, for m 2 Zr , X X ihn,‚.m/i e Un D Un V‚.m/=2. jnjl jnjl

Consider the following partial sums (we apply below Lemma 12.4.11). X X ihn,‚.m/i ihn,‚.m/i e .UmMf Un/g D e .Umf /.Ung/ jmnjk,jnjl jmjk,jnjl X X ihn,‚.m/i D .Umf/ e Ung jmjk jnjl X X D .Umf/ Un V‚.m/=2g . jmjk jnjl 404 Chapter 12 Residues and Integrals in Noncommutative Geometry

When l !1, the latter sum converges (in the norm topology) in L2.X/ to the function X N bk :D .Umf /.V‚.m/=2g/, k 2 . jmjk

When k !1, the latter sum converges uniformly in L2.X/ by Lemma 12.4.10. Indeed, X X .Umf /.V‚.m/=2g/ kUmf k1kV‚.m/=2gk2 ! 0, 2 k1

d=2 d=2 Int!.L‚.f // D Tr!.L‚.f /.1 g/ / D Tr!.L‚.Mf .1 g/ // d=2 D Tr!.Mf .1 g/ / D Int!.Mf /.

The assertion of (b) now follows from Theorem 11.7.10, recalling our use of a scaling d=2 factor so that Tr!..1 g / / D 1.

d=2 Proof of Corollary 12.4.4. The operator A.1 g / 2 L.L2.X// is Hodge– Laplacian modulated by Lemma 11.2.2 and Lemma 11.2.8. By Lemma 12.4.9, we d=2 obtain a Hodge–Laplacian modulated operator L‚.A.1 g/ / such that X ihn,‚.m/i d=2 d=2 e Um.A.1 g / /Un ! L‚.A.1 g/ / jmj,jnjk in L2.L2.X// when k !1. r Since Um, m 2 Z , commutes with g , it follows that the square partial sums for L‚.A/ in (12.6) can be rewritten as X ihn,‚.m/i d=2 d=2 e Um.A.1 g / /Un .1 g/ . (12.8) jmj,jnjk

d d=2 If f 2 H .X/,then.1 g / f 2 L2.X/ and partial sums in (12.8) converge on d f . This proves strong convergence of (12.6) in the space L.H .X/, L2.X//. That

d=2 d=2 Int!.L‚.A// D Tr!.L‚.A.1 g / // D Tr!.A.1 g / / D Int!.A/, follows from Theorem 12.4.2. Section 12.5 Example of the Noncommutative Torus 405

12.5 Example of the Noncommutative Torus

The simplest version of an isospectral deformation is the noncommutative torus. In this section, to illustrate the formulas of the preceding sections, we examine concrete formulas for the noncommutative torus. Let 2 H :D M2.L2.T // denote the Hilbert space of 2 2 matrices of square integrable functions on the torus T2 D R2=Z2. We can identify, without loss, the square integrable functions on the 2 torus with L2.Œ0, 1 / (the boundary, as a set of measure zero, is irrelevant with regard to equivalence classes of functions up to measure zero). We identify the smooth func- 1 2 1 2 tions C .T / on the torus with those f 2 C .Œ0, 1 / such that f.s1,0/ D f.s1,1/ and f.0, s2/ D f.1, s2/, s1, s2 2 Œ0, 1. Deﬁne the Hodge–Dirac operator

1 2 1 2 D : M2.C .T // ! M2.C .T // by @ @ i 0 @s i @s D :D p 1 2 , s1, s2 2 Œ0, 1. 2 @ C i @ 0 @s1 @s2

1 2 Here D acts on M2.C .T // by the same formula as matrix multiplication and Dirichlet boundary conditions are assumed. The unbounded Fredholm module of interest for the noncommutative torus is .H , D/. As noted in the last section, to look at the noncommutative integral we can work, without loss, with the Laplacian on the 2 2 Hilbert space L2.T / since D D 122. The only difference is a factor of 4 in the multiplicity of the eigenvalues, since each eigenspace of D2 is the corresponding eigenspace of tensored by the four-dimensional Hilbert space M2.C/. Note that the Hodge-Dirac operator acts on smooth sections of the exterior bundle, represented 1 2 as M2.C .T //, and that it is different from the Dirac operator acting on smooth sections of the spinor bundle, which would be represented by C 1.T2/ ˚ C 1.T2/, [152]. The torus T2 can be considered as the joint spectrum of the functions

2is1 2is2 2 u.s1, s2/ :D e , v.s1, s2/ :D e , s D .s1, s2/ 2 T ,

2 2 acting as multiplication operators on L2.T /. The functions u and v generate L2.T / in the sense of Fourier series expansions, X O n1 n2 2 f D f.n1, n2/u v , f 2 L2.T /. 2 nD.n1,n2/2Z 406 Chapter 12 Residues and Integrals in Noncommutative Geometry

2 We recall that the Lebesgue integral of f 2 L2.T / is given by the ﬁrst Fourier coef- ﬁcient, Z f.s/ds D f.O 0/. T2

The Laplacian : C 1.T2/ ! C 1.T2/,isgivenby 1 @2 @2 D 2 C 2 4 @s1 @s2

2 2 (with Dirichlet boundary conditions) and fn Djnj fn, n 2 Z , where the functions

n1 n2 2 2 2 fn :D u v 2 L .T /, n D .n1, n2/ 2 Z

2 2 form an orthonormal basis of L2.T /. We choose an enumeration of Z by choosing a function k ! n.k/, k 0, such that jn.k/j, k 0, is an increasing sequence. Denote

ek :D fn.k/, k 0, and p k :D jn.k/j, k 0.

2 1 Then ek D kek and fekgkD0 forms an ordered eigenbasis for with [38],

2 1 hki .1 C k/ .

As in Sections 10.3 and 11.6, the extension of the operator is a positive unbounded T2 operator on L2. / with compact resolvent. Thep noncommutative integral associated 2 to the unbounded Fredholm module .L2.T /, / is deﬁned for any bounded op- 2 erator on the Hilbert space L2.T / by

1 2 Int!.A/ :D Tr!.A.1 / /, A 2 L.L2.T // where Tr! is a Dixmier trace. We have normalized the Laplacian in this section so 1 that Tr!..1 / / D 1. Section 11.7, in particular Theorem 11.7.10, indicates that the noncommutative in- 2 2 tegral is an extension of the Lebesgue integral from L1.T / to all of L.L2.T //.The spectral formulas in Theorem 12.3.1 provide the same conclusion in a more direct fashion.

Theorem 12.5.1. For every dilation invariant extended limit on l1, Z 2 Int!.Mf / D f.s/ds, f 2 L1.T /, T2 Section 12.5 Example of the Noncommutative Torus 407

2 where Mf 2 L.L2.T // denotes the action of an essentially bounded function f on square integrable functions by pointwise multiplication.

Proof. Observe that ekek D 1, k 0whereek is an element of the ordered basis of the Laplacian deﬁned above. Then Z Z

hMf ek, ekiD f.s/ek.s/ek.s/ds D f.s/ds, k 0. T2 T2

Trivially then, Z

Int!.Mf / D lim hMf ek, ekiD f.s/ds, k!1 T2 from Theorem 12.3.1.

The smooth functions u and v commute and this property represents commuting coordinates for the torus. The idea behind the noncommutative torus is to deform the pointwise product of the algebra C 1.T2/ so that u and v do not commute for this new product (thus representing “noncommuting” coordinates). We set up the isospectral deformation as in the last section. Deﬁne the action of the torus on itself by

2 ˛s.t/ :D .s1 t1 mod 1, s2 t2 mod 1/, s D .s1, s2/, t D .t1, t2/ 2 T .

This deﬁnes a strongly continuous representation of the torus into the unitary group of 2 L2.T / by

2 2 .Vsf /.t/ :D f.˛s .t// D f.s C t/, f 2 L2.T /, s, t 2 T .

Deﬁne, Z 2ihm,si 2 Um :D e Vsds, m 2 Z . T2

Let ƒ denote the additive group of skew symmetric matrices in M2.T/,whereT D 2 R=Z is the 1-torus. If A 2 L.L2.T // and ‚ 2 ƒ, deﬁne, formally, X ihn,‚.m/i L‚.A/ :D e UmAUn, m,n2Z2 where the same considerations must be made about the convergence of the square partial sums, as in Section 12.4. The map L‚ is not an algebra homomorphism. When 2 2 A, B 2 L.L2.T // such that L‚.A/, L‚.B/ 2 L.L2.T //, the product

2 L‚.A/L‚.B/, A, B 2 L.L2.T // 408 Chapter 12 Residues and Integrals in Noncommutative Geometry

deforms the product L0.A/L0.B/ D AB. The main result of this section is that the deformation of the smooth functions on T2 forms a noncommutative algebra called the noncommutative torus, and that the noncommutative integral is a trace on the noncommutative torus. L T2 Note that, as in the last section, we shorten L‚.Mf /, Mf 2 .L2. //, f 2 1 T2 1 T2 C . /,toL‚.f / henceforth. Then L0.f / D Mf , f 2 C . /.

Deﬁnition 12.5.2. The noncommutative torus N‚ is the uniform closure of the set of bounded operators generated by L‚.u/ and L‚.v/.

It can be seen below that, if a 2 N‚,then X n1 n2 a D a.n1,n2/L‚.u/ L‚.v/ , nD.n1,n2/ where the sum converges in the uniform operator topology. A skew-symmetric matrix ‚ 2 ƒ is described by a single parameter, 0 ‚ :D , 2 T. 0

The next result shows that the deformation, ‚ 6D 0, corresponds to a Weyl quantization of the ordinary torus, by turning the commuting generators u and v of the torus into non-commuting generators L‚.u/ and L‚.v/ that satisfy the Weyl relations. When 2 ‚ D 0 the algebra N0 is the commutative C -algebra of continuous functions C.T /, with pointwise product.

Theorem 12.5.3. If ‚ 6D 0, then the noncommutative torus N‚ is a noncommutative 2 C -subalgebra of L.L2.T //. In particular, L‚.u/ and L‚.v/ are unitaries such that

2i L‚.u/L‚.v/ D e L‚.v/L‚.u/.

1 2 1 2 If C .T / denotes the smooth functions on the torus, then N‚ D L‚.C .T //. The proof of the theorem is given below. As a consequence of Theorem 12.4.2, the noncommutative integral restricted to the noncommutative torus is deﬁned densely by the Lebesgue integral.

Corollary 12.5.4. All operators in the C -algebra N‚ are q-measurable and Int is a hypertrace on N‚ such that Z 1 2 Int.L‚.f // D f.s/ds, f 2 C .T /. T2

We prove the theorem and the corollary with the aid of the following lemma. Section 12.5 Example of the Noncommutative Torus 409

Lemma 12.5.5. If u and v are the generators of the torus and 0 ‚ D , 2 T, 0 then 2i L‚.u/L‚.v/ D e L‚.v/L‚.u/, and

k1 k2 k1 k2 2 L‚.u v / D L0.u v /V 1 , k D .k1, k2/ 2 Z , 2 ‚.k/ and

k1 k2 ick ,k k1 k2 2 L‚.u/ L‚.v/ D e 1 2 L‚.u v /, k D .k1, k2/ 2 Z ,

Z for coefﬁcients ck1 ,k2 2 . Proof. We use Corollary 12.4.4. Observe that (with the notation from that corollary) k k 2ihm,ri 2 2 .Umu 1 v 2 /.r/ D e ı0.m k/, m 2 Z , r 2 T ,whereı0 is the Kronecker delta. From Corollary 12.4.4, X k k k k 1 2 k k 1 2 L‚.u v / D M m 1 2 V 1 D L0.u v /V 1 . (12.9) U u v 2 ‚.m/ 2 ‚.k/ m2Z2

The second assertion is shown. In particular,

L‚.u/ D L0.u/V 1 , L‚.v/ D L0.v/V 1 . 2 ‚.1,0/ 2 ‚.0,1/ Then

L‚.u/L‚.v/ D L0.u/V 1 L0.v/V 1 2 ‚.1,0/ 2 ‚.0,1/ D L0.u/V 1 L0.v/V 1 V 1 2 ‚.1,0/ 2 ‚.1,0/ 2 ‚.1,1/ i i D e L0.uv/V 1 D e L‚.uv/ 2 ‚.1,1/ where we have used (12.9) and the fact that

1 2i.s2 / i V 1 L0.v/V 1 D e 2 D e L0.v/. 2 ‚.1,0/ 2 ‚.1,0/ Similarly,

i L‚.v/L‚.u/ D L0.v/V 1 L0.u/V 1 D e L0.uv/V 1 2 ‚.0,1/ 2 ‚.1,0/ 2 ‚.1,1/ i D e L‚.uv/. 410 Chapter 12 Residues and Integrals in Noncommutative Geometry

The ﬁrst assertion is now shown since

i 2i L‚.u/L‚.v/ D e L‚.uv/ D e L‚.v/L‚.u/.

From similar workings, there exist coefﬁcients cp 2 Z, p 2 Z, such that

k1 ick k1 L‚.u/ D e 1 L0.u /V 1 2 ‚.k1,0/ and

k2 ick k2 L‚.v/ D e 2 L0.v /V 1 . 2 ‚.0,k2/ Hence,

k1 k2 i.ck ck / k1 k2 L‚.u/ L‚.v/ D e 1 2 L0.u /V 1 L0.v /V 1 2 ‚.k1,0/ 2 ‚.0,k2/ i.ck ck / k1 k2 D e 1 2 L0.u /V 1 L0.v /V 1 V 1 2 ‚.k1,0/ 2 ‚.k1,0/ 2 ‚.k/ i.ck ck k2/ k1 k2 ick ,k k1 k2 D e 1 2 L0.u v /V 1 D e 1 2 L‚.u v / 2 ‚.k/ for some coefﬁcient ck1 ,k2 . Proof of Theorem 12.5.3. It is clear from Lemma 12.5.5 that if ‚ 6D 0, i.e. 2 Œ0, 1/ mod 1 is not equal to 0, that L‚.u/ and L‚.v/ are unitary operators and they do not commute. Hence, N‚ is a noncommutative algebra of bounded operators. That it is a C -algebra follows. Take f 2 C 1.T2/ and X f D f.k/uO k1 vk2 . 2 kD.k1,k2/2Z

Lemma 12.5.5 shows that X ick ,k k1 k2 L‚.f / D e 1 2 f.k/LO ‚.u/ L‚.v/ . 2 kD.k1,k2/2Z

The convergence of partial sums is uniform, since X X ick ,k k1 k2 L‚.f / e 1 2 f.k/LO ‚.u/ L‚.v/ jf.k/O j!0 1 k2Œn,n2 k62Œn,n2 as n !1, where the converge to zero is due to the fact that f is smooth and hence fO is a sequence of rapid decrease. Thus, L‚.f / 2 N‚. By choosing those f such ick ,k 1 2 that e 1 2 f.k/O 2 c00, clearly L‚.C .T // is dense in N‚. The assertion is shown. Section 12.6 Classical Limits 411

1 2 Proof of Corollary 12.5.4. If a :D L‚.f /,wheref 2 C .T /, then, by Corol- lary 12.4.4 we have that a is q-measurable and that

Int!.a/ D Int!.L0.f // D f.O 0/.

1 2 The last equality is from Theorem 12.5.1. Since L‚.C .T // is norm dense in N‚ and Int! is a singular state (continuous in the uniform operator topology), it follows N that all operators in ‚ are q-measurable. p To prove that Int is a hypertrace we show that Œ , a is bounded for a dense N subset of ‚, and then apply Corollary 12.3.5(b). Set a :D L‚.fp / for a smooth function f . By Theorem 12.5.3 such elements are dense in N‚.SinceŒ , Vs D 0 for all s 2 T2,then p X p ick ,k O k1 k2 Œ , L‚.f / D e 1 2 f.k/Œ , L0.u v /V 1 2 ‚.k/ kDX.k1,k2/ ick ,k O k1 k2 D e 1 2 f.k/jkjL0.u v /V 1 . 2 ‚.k/ kD.k1,k2/

Then p X kŒ , L‚.f /k1 jf.k/O jjkj < 1

kD.k1,k2/ since fO is a sequence of rapid decrease when f is smooth. The hypertrace property 1 2 follows from Theorem 12.3.4 since L‚.C .T // is dense in N‚.

12.6 Classical Limits

Singular traces are fundamental objects in functional analysis. They are all the traces besides the extension of the matrix trace, and are as fundamental as the ideal structure in the compact operators discovered by J. von Neumann and his students R. Schatten and J. W. Calkin. Singular traces, in the guise of the noncommutative residue, are also fundamental to Riemannian geometry. As functional analysis and Riemannian geometry play such pivotal roles in the physical theories of quantum mechanics and general relativity, what role do singular traces play in physical theories? Answers to this question take us further afield, and deeper into Alain Connes’ quantum calculus. This is the task of other monographs. In this section, the last section of this book, we reflect on the formulas proved in previous chapters and their connection to classical limits. Connes, in the introduction to the paper which contained what we now call Connes’ Trace Theorem, posited the Dixmier trace as a replacement for taking the “classical limit”. Links between the noncommutative integral based on the Dixmier trace (a 412 Chapter 12 Residues and Integrals in Noncommutative Geometry state vanishing on the finite rank projections, where the projections represent, essentially, quanta), limits of Gibbs states (the high temperature limit), the limit at infinity of the expectation values (the large quantum number limit), and the invariance of the noncommutative integral under isospectral deformation (a strict deformation quantization [144]), provide remarkable substance to this idea. The link between the formulations of the noncommutative integral as the high temperature limit and the large quantum number limit is in Theorem 12.6.1 below. It combines the observations from Chapter 8, Chapter 9, and Section 12.3. Invariance under isospectral deformation was shown in Section 12.4. We review the ideas of the high temperature limit and the large quantum number limit before stating the theorem. As in the previous sections, let .H , D/ be an unbounded Fredholm module of dimension d.Thatis,H is a separable Hilbert space and D :Dom.D/ ! H is unbounded self-adjoint operator, with compact resolvent, such that

d 2 d=2 hDi :D .1 C D / 2 L1,1.

1 An ordered eigenbasis for D is a basis fengnD0 of H such that Den D nen, n 0, and jnj, n 0, is an increasing sequence. First let us consider the Gibbs state on an operator A 2 L.H /,whereˇ acts as a 1 2 scale for temperature. Set g.t/ :D .1 C t 2/.dC1/=2eˇ t , t 0, which is a bounded d1 function, kgk1 < 1.Sinceg.D/ is bounded and hDi 2 L1 is trace class then, using the functional calculus of self-adjoint operators,

ˇ 1D2 d1 e D g.D/hDi 2 L1, ˇ>0. L We deﬁne the states Gˇ 2 .H / , ˇ>0, P 1 2 1 1 2 Tr.Aeˇ D / hAe , e ieˇ n nDP0 n n L Gˇ .A/ :D 1 2 D 2 , A 2 .H /. ˇ D 1 ˇ 1n Tr.e / nD0 e The terms

1 2 1 ˇ n P 1 2 e , n 0, 1 ˇ j j D0 e represent the probability of being in the eigenstate en, n 0. What happens to these states as ˇ !1is called the high temperature limit [116]. Conceptually, as ˇ !1 the probability of being in any eigenstate is becoming uniform. With higher energy, all eigenstates are becoming saturated and, essentially, the quantum states are becoming indistinguishable. The states Gˇ , ˇ>0, may not have a limit as ˇ !1. A limit can be obtained in an extended sense by applying an extended limit on L1.RC/. Theorem 12.6.1 shows that for certain extended limits, the limit of the Gibbs states is the noncommutative integral Section 12.6 Classical Limits 413 in noncommutative geometry, and that an operator A 2 L.H / is q-measurable if the states Gˇ .A/, ˇ>0, logarithmically converge as ˇ !1. The expectation 1 L A !fhAen, enignD0 , .H / ! l1, associates to a self-adjoint operator A, representing an observable, the expected observable value for the eigenstate en, n 0. The correspondence principle in quantum mechanics is that the “classical limit” is the large quantum number limit of the quan- 1 tum system, that is, the limit as n !1when fengnD0 is an ordered eigenbasis. The quantum harmonic oscillator is an example of the principle [244]. Conceptually, the 1 “limit at inﬁnity” of the sequence fhAen, enignD0 represents the observable value in the classical limit. The sequence, generally, does not converge. A limit can be obtained in an extended sense by applying an extended limit on l1. Theorem 12.3.1 shows that, for extended limits composed with the logarithmic mean, the limit of the expectation values is the noncommutative integral in noncommutative geometry. The next theorem mathematically unites the “correspondence principle” and the “high temperature limit” to each other and to Connes’ noncommutative integral. We restrict an extended limit ! on L1.0, 1/ to an extended limit on l1 by X1 1 !.x/ :D ! xnŒn,nC1/ , x DfxngnD0 2 l1. nD0

1 Recall that an orthonormal basis fengnD0 is an ordered eigenbasis of D if Den D nen, n 0, and jnj, n 0, is an increasing sequence. Recall also that an unbounded d Fredholm module .H , D/satisfies the Weyl asymptotic condition if jnj n, n 0. In the statement below M denotes the discrete logarithmic means (see the definition before Theorem 12.1.2) and M denotes the continuous logarithmic means (see the definition before Theorem 9.2.1).

Theorem 12.6.1. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl asymptotic condition. If A 2 L.H / then ˇ 1D2 1 Tr.Ae / Int!.A/ D ! ı M.fhAej , ej ig D / D ! ı M , j 0 Tr.eˇ 1D2 /

1 for any ordered eigenbasis fengnD0 of D.Here! is an extended dilation invariant limit on L1.0, 1/ such that ! ı log is also dilation invariant, and the two formulas on the left involve the restriction of ! to an extended limit on l1. Connes’ notion of measurability in noncommutative geometry coincides with the sequence of expectation values and the Gibbs states converging logarithmically. 414 Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.6.2. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl 1 asymptotic condition and fengnD0 be an ordered eigenbasis of D. An operator A 2 L.H / is q-measurable if and only if the following values exist and coincide ˇ 1D2 1 Tr.Ae / Int.A/ D lim ıM.fhAej , ej ig D / D lim ıM . j 0 Tr.eˇ 1D2 /

Theorem 12.6.1 and Theorem 12.6.2 are proved below. First we require a preliminary lemma.

1 Lemma 12.6.3. If 0 A 2 L1,1 is such that .n, A/ .1 C n/ , n 0,then 1 1 q 1 ˇ q Tr.eˇ A / D .1 C o.1// 1 C , ˇ !1, q>0. (12.10) q

Proof. Fix >0 and select n 1 sufﬁciently large so that k.k, A/ 2 .1 ,1C / for k n. Then, X1 X1 1 q q 1 q 1 q q eˇ .1/ k C O.1/ Tr.eˇ A / eˇ .1C/ k C O.1/. kD1 kD1

Therefore, Z Z 1 1 1 q q 1 q 1 q q eˇ .1/ s dsCO.1/ Tr.eˇ A / eˇ .1C/ s dsCO.1/. 0 0 Computing the integrals in terms of the -function, we obtain that 1 1 q 1 .1 /ˇ1=q 1 C C O.1/ Tr.eˇ A / .1 C /ˇ1=q 1 C C O.1/. q q

Since is arbitrarily small, the assertion follows.

Lemma 12.6.4. Let 0 A 2 M1,1 be positive and Dixmier measurable and let B 2 L.H /. The operator AB is Dixmier measurable if and only if the following limit exists for some q>0, 1 q lim M Tr.Be.ˇA/ / .t/. t!1 ˇ

Proof. We use results and terminology from Chapters 6, 7, 8 and 9. Set D Tr. Suppose that the limit exists. Setting f.t/ :D exp.t q /, t>0, and ﬁxed q> 0, we infer that the heat kernel functionals !,B,f .A/ take the same value for every dilation invariant extended limit !. It follows from Corollary 8.5.2 that ! .AB/ does Section 12.6 Classical Limits 415 not depend on !. By Theorem 8.3.6 and Theorem 6.4.1, the set of Dixmier traces and the set of heat kernel formulas coincide. Hence, !.AB/ does not depend on ! and, therefore, AB is Dixmier measurable. Conversely, let AB be Dixmier measurable. Since A is self-adjoint, it follows that

!.A.

It follows that A.

Hence, for B 2 L.H / and any extended limit ! on L1.RC/, ˇ 1D2 Tr.Be / 1 d=2 ˇ 1D2 .! ı M/ D .! ı M /.ˇ Tr.Be //. Tr.eˇ 1D2 / .1 C d=2/

By assumption, ! ı Ps D ! for every s>0wherePs, s>0, denotes the exponentiation operators. Applying Pd=2 and using the fact that M ı Ps D Ps ı M ,weinfer that ˇ 1D2 Tr.Be / 1 1 .ˇ hDid /2=d .! ı M/ D .! ı M /.ˇ Tr.Be //. Tr.eˇ 1D2 / .1 C d=2/ 416 Chapter 12 Residues and Integrals in Noncommutative Geometry

It follows from Corollary 8.5.2 that ˇ 1D2 Tr.Be / d .! ı M/ D !.BhDi /. Tr.eˇ 1D2 /

Since ! D ! ı Ps for s>0, it follows from Theorem 8.2.9 that ! D !.By Theorem 6.4.4 and comments in Section 10.1, ˇ 1D2 Tr.Be / d .! ı M/ D Tr!.BhDi / Tr.eˇ 1D2 / when ! is restricted to l1.That

d 1 Tr!.BhDi / D ! ı M.fhAej , ej igj D0/ is the statement of Theorem 12.3.1 (a).

Proof of Theorem 12.6.2. Let B 2 L.H /. Using equation (12.10), the limit

1 2 Tr.Beˇ D / .lim ıM/ (12.11) Tr.eˇ 1D2 / exists if and only if the limit

1 2 .lim ıM /.ˇd=2Tr.Beˇ D // exists. Applying Pd=2 and using the fact that M ı Ps D Ps ı M , we infer that these limits exist if and only if

d 2=d .lim ıM /.ˇ1Tr.Be.ˇ hDi / // exists. This limit exists if and only if BhDid is measurable by applying Lemma 12.6.4 to the operator A DhDid . d 1 That BhDi is measurable if and only if lim ıM.fhBej , ej igj D0/ exists is the statement of Theorem 12.3.1 (b).

12.7 Notes

Residues and Integrals in Noncommutative Geometry Spectraltriples were introducedby A. Connes in [49], they have also been calleda K-cycle over a C -algebra [48, p. 546], and an unbounded Fredholm module over a C -algebra [7,47]. The operator D is a generalizationof a Dirac type operator [146]. The Connes–Moscovici approach to the pseudo-differential calculus of spectral triples was introduced in [54] and [49]. Section 12.7 Notes 417

We have not needed to discuss the differential side of the quantum calculus in this book. We have only mentioned that the algebra of all operators A 2 L.H / such that ŒhDi, A 2 L.H / considered in Corollary 12.3.5 contains the set Op0 of the Connes–Moscovici pseudo- differential calculus, [54]. We refer the reader to [30] and [31] for semiﬁnite version of this calculus. In [48, §VI], [49,50,54], Connes introduced his quantized calculus with the compact oper- 1 2 1 ator hDi :D .1 C D / 2 as the analog of an inﬁnitesimal length element ds. Following the integration of smooth functions in the pseudo-differential calculus using Wodzicki’s noncommutative residue, the integral of A 2 N ,here.N , H , D/is a spectral triple, was expressed by Connes as the formula [45, pp. 157–158], [48, p. 545]

d Tr!.AhDi /, A 2 N . (12.12)

d The requirement for hDi 2 L1,1 identifies the (finite) dimension of the triple. A more refined concept of dimension of spectral triples is given by their dimension spectrum, see [54, II.1] and [112]. The existence of a Gibbs state,

ˇD2 e 2 L1, ˇ>0 , (12.13) was suggested as a replacement dimension condition for infinite dimensional triples [46, 47], and surfaces in quantum field theory applications of noncommutative geometry [47, §7], [117], [48, IV.9]. An alternative integral based upon Gibbs states was proposed for the condition (12.13) at ([37], p. 208) and in ([94], §2.1.3) (for brevity we use the same notation as the referenced texts): 1 2 Tr.Aer D / A :D ` , A 2 N , (12.14) Tr.er1D2 / ` where ` was an unspecified limiting procedure at infinity. That the functional in (12.14) yields the same value as a Dixmier trace for a finite dimensional triple was not shown in ([37], p. 208) or ([94], §2.1.3) outside the assumption that A is q-measurable. Hardy, in [108, §3.8], illustrated the difference between the Cesàro and logarithmic means. In particular, the logarithmic means are less trivial (more useful) than the Cesàro means, meaning that moreP sequences converge logarithmically than converge arithmetically. If p D 1 fpngnD0 > 0and pn diverges, Hardy defined the means .N, p/ : l1 ! l1, P n x p Pj D0 j j 1 ..N, p/.x//n :D n , x Dfxj gj D0 2 l1, n 0. j D0 pj

1 The CesàroP means are the .N, 1/ means and the logarithmic means are the .N, .1 C n/ / means. As pn diverges more quickly the means .N, p/becomes more trivial, until reaching the point that only convergent sequences end up being .N, p/-convergent [108, Theorem 15]. An exact statement for when a logarithmically convergent sequence is a convergent sequence 1 is known [163, Theorem 5]: a logarithmically convergent sequence x DfxngnD0 2 l1 is a convergent sequence if and only if

ˇ Œnr ˇ ˇ 1 X x x ˇ lim lim sup ˇ k n ˇ D 0. C ˇ r ˇ r!1 n!1 .Œn n/ log.1 C n/ k kDnC1 418 Chapter 12 Residues and Integrals in Noncommutative Geometry

Here Œnr denotes the ceiling of nr , n 1, r>1. The more testable Tauberian criteria of slowly oscillating real sequences was shown by B. Kwee in [142]. Equivalent criteria to slowly oscillating is given in [163]. Observations betweenthe noncommutativeintegral and the logarithmic mean of expectation values were ﬁrst noted in [154]. For the notion of quantum ergodicitysee [43,74,204,220,266]. The expectation value approach to traces has also been used in Pietsch’ work on traces on Banach ideals [184], as well as in [126], which extended [154]. Connes [47,48] ﬁrst noted the hypertrace property of the noncommutative integral for spectral triples. The proof we have given follows [42] and observations in [33].

Isospectral Deformations The Myers–Steenrod Theorem, [167], proves that the isometry group Iso.X/ of a compact manifold X is a compact Lie group. The Lie algebra of Iso.X/, i.e. the infinitesimal generators of isometries, is composed of the Killing vector fields on X. If the dimension of X is d ,asa finite dimensional smooth manifold the dimension of Iso.X/ is less than or equal to d.d C 1/=2 [134]. From the theory of compact Lie groups, the maximal connected, commutative, subgroups of a compact Lie group all have the same dimension r, and are all isomorphic to the r-torus [219]. These subgroups are called maximal torii and the value r is called the rank of the compact Lie group. Therefore, closed manifolds whose isometry group Iso.X/ has rank 2 admit smooth representations of the torus, and are available for isospectral deformation [53]. Killing fields relate to metric preserving symmetries of the manifold. Two central Killing fields are enough to guarantee a rank 2 for the isometry group [219, Corollary 5.13]. Conversely, if it is known that the compact Lie group Tr acts smoothly on a compact manifold X,then there is a metric g on X such that Tr Iso.X/ [202, Corollary 6.3]. That is, one can always choose a metric g such that the smooth action of a torus Tr can be represented by unitaries that commute with the Laplace–Beltrami operator g . Connes, Landi, and Dubios-Violette [52,53] introduced the notion of a noncommutativegeometry associated to an isospectral deformation of a closed Riemannian manifold. It involves the discrete version of the Moyal product [101, 164, 192]. Bruno Iochum, Victor Gayral and their collaborators [95, 96], considered the continuous Moyal product and isospectral deformations of non-compact manifolds. Their paper [96] showed the result in Corollary 12.4.3 for compact (or, under certain conditions, non-compact) Riemannian manifolds. That if A is Hilbert–Schmidt, then the deformation L‚.A/ is Hilbert–Schmidt was shown in [96], as was the property that Tr.L‚.A/p/ D Tr.Ap/ for a spectral projection p of the Hodge– Laplacian.The method of proof of Theorem 12.4.3 in Section 12.4 is different from that in [96], which was based on -function estimates. -function estimates cannot show that, as in Theo- rem 12.4.2, isospectral deformation of modulated operators is trace invariant for every trace on L1,1. The exposition of isospectral deformations given in Sections 12.4 and 12.5 is close to [261]. The noncommutative torus originated, as an example of a noncommutative geometry, with [44,48,50,55] and [246]. It appears now in innumerable papers, particularly in connection with String Theory, e.g. [51, 137]. The noncommutative torus has other, equivalent, presentations besides as an isospectral deformation. Usually the emphasis is on irrational values for the parameter [191]. Connes used a -function residue to recover the trace on the noncommutative torus in [47]. The computation of the -function of the noncommutative torus can be found in [87]. Section 12.7 Notes 419

Classical Limits For deeper aspects of Connes’ quantum calculus, such as the local-index formula, and recovery of the Einstein–Hilbert action, see [30,31,48,50,54,131]. Connes’ posited the Dixmier trace as a replacement for taking the classical limit in [45]. There is no completely deﬁnitive approach to taking the “classical limit”. The high temperature limit is an approach in statistical mechanics [116], the correspondence principle goes back to the time of Bohr, as does the Ehrenfest notion of expected values [16, 145], mathematically there is strict quantization of Poisson structures [144]. The interesting conceptual feature of the invariance of the noncommutative integral under isospectral deformation is that it, the noncommutative integral, is not so much a classical limit, but rather the stronger property of being an invariant of quantization. Appendix A Operator Results

A.1 Matrix Results

The following results on matrices were used at speciﬁc places in the text, especially for the results on quasi-nilpotent compact operators in Chapter 5. In Section 5.4, if u : C ! R is continuous we deﬁned a function uO : Mn.C/ ! R by setting X u.A/O :D u./, A 2 Mn.C/. 2.A/

Set

logC.x/ :D maxflog.jxj/,0g, x 2 R. b b Lemma A.1.1. For every operator A 2 Mn.C/, we have logC.A/ logC.jAj/. Proof. Let

N1 :Djf 2 .A/ : jj > 1gj, N2 :Djf 2 .jAj/ : jj > 1gj.

It follows from Lemma 1.1.20 that

Y YN1 YN1 jjD j.k, A/j .k, A/. 2.A/,jj>1 kD0 kD0

If N2 N1,then.k, A/ > 1 for every k 2 .N1, N2. Therefore,

YN1 YN2 Y .k, A/ .k, A/ D jj.(A.1) kD0 kD0 2.jAj/,jj>1

If N2

YN1 YN2 Y .k, A/ .k, A/ D jj.(A.2) kD0 kD0 2.jAj/,jj>1

Combining (A.1) and (A.2), we conclude the proof. Section A.1 Matrix Results 421

In Section 2.3 we deﬁned the distribution function of an operator L.H / (or a matrix),

nA.s/ :D Tr.EjAj.s, 1//, s 0.

Lemma A.1.2. For every normal operator A 2 Mn.C/ and every ˛ 2 C such that j˛j1, we have ˇ ˇ ˇ X X ˇ ˇ ˇ ˇ ˛ ˇ nA.1/. 2.˛A/,jj>1 2.A/,jj>1

Proof. It is clear that ˇ ˇ ˇ X X ˇ ˇ ˇ ˇ ˛ ˇ 2.˛A/,jj>1 2ˇ.A/,jj>1 ˇ ˇ ˇ ˇ X ˇ ˇ X ˇ ˇ ˇ ˇ ˇ D ˇ ˛ˇ ˇ 1ˇ D nA.1/. 2.A/,11

Lemma A.1.3. Let A1, A2, A3 2 Mn.C/ be normal operators such that A1 C A2 C A3 D 0. We have ˇ ˇ ˇ X3 X ˇ ˇ ˇ ˇ ˇ 2.nA1 .1/ C nA2 .1/ C nA3 .1//. iD1 2.Ai /,jj>1

Proof. Let pi D EjAi j.1, 1/, i D 1, 2, 3. Since every Ai is normal, it follows that C pi D EAi .fz 2 , jzj > 1g/. Hence, X D Tr.pi Ai pi /, 8i D 1, 2, 3. 2.Ai /,jj>1

Set _ p D pi . iD1,2,3

We have

Tr.pi Ai pi / D Tr.pAi p/ Tr..p pi /Ai .p pi //.

Since A1 C A2 C A3 D 0, it follows that

X3 X3 Tr.pi Ai pi / D Tr..p pi /Ai .p pi //. iD1 iD1 422 Appendix A Operator Results

We have

k.p pi /Ai .p pi /k1 k.1 pi /Ai .1 pi /k1 1 and

rank..p pi /Ai .p pi // Tr.p pi / D Tr.p/ Tr.pi /.

It follows from the obvious inequality jTr.A/jkAk1 rank.A/ that

Tr..p pi /Ai .p pi // Tr.p/ Tr.pi /.

Therefore, ˇ ˇ ˇ X3 X ˇ X3 ˇ ˇ ˇ ˇ Tr.p/ Tr.pi /. iD1 2.Ai /,jj>1 iD1

It is clear that Tr.pi / D nAi .1/ for i D 1, 2, 3 and

Tr.p/ nA1 .1/ C nA2 .1/ C nA3 .1/.

The assertion follows immediately.

A.2 Operator Inequalities

In this section .M, / denotes a von Neumann algebra M equipped with a ﬁxed normal semiﬁnite trace . The results presented below can be found in [18, 23, 32].

Lemma A.2.1. Let A, C 2 M be self-adjoint operators such that A C . It follows that .AC/ .CC/.

Proof. There exists projection p 2 M such that CC D pCp.Wehavep.AC/p 0 and, therefore, pAp CC. Hence, pACp pAp C CC. Thus, .CC/ .pAp C CC/ .pACp/ .AC/.

Lemma A.2.2. Let A, B 2 M be positive operators. We have

1=2 1=2 1=2 1=2 .B .A 1/CB / ..B AB 1/C/ provided that B 1.

Proof.LetC D 1 C .A 1/C.WehaveA C and, therefore,

B1=2AB1=2 1 B1=2CB1=2 1. Section A.2 Operator Inequalities 423

It follows from Lemma A.2.1 that

1=2 1=2 1=2 1=2 ..B AB 1/C/ ..B CB 1/C/.

We have B 1 and, therefore,

B1=2CB1=2 1 B1=2.C 1/B1=2.

It follows from Lemma A.2.1 that

1=2 1=2 1=2 1=2 1=2 1=2 ..B CB 1/C/ .B .C 1/B / D .B .A 1/CB /.

Lemma A.2.3. Let A, B 2 M be positive operators. For every convex continuous increasing function f , we have

.B1=2f.A/B1=2 / .f.B1=2AB1=2// provided that B 1.

Proof. First, consider the case when supp.A/ 2 L1.M, /.Wehave

.supp.B1=2AB1=2//, .supp.B1=2f.A/B1=2// .supp.A//.

On the interval Œ0, kAk1, the function f admits a uniform approximation with piecewise-linear convex increasing continuous functions. Hence, for every >0, there exists Xn g : t ! ˛k.t ˇk/C, kD1 such that ˛k, ˇk 0, 1 k n,and

f.t/ g.t/ f.t/C , 80 t kAk1.

By Lemma A.2.2, we have

.B1=2g.A/B1=2/ .g.B1=2AB1=2//.

It follows that

.B1=2f.A/B1=2/ .f.B1=2AB1=2// .supp.A//.

Since >0 is arbitrarily small, the assertion is proved for every operator A with supp.A/ 2 L1.M, /. 424 Appendix A Operator Results

In the general case, we have 1=2 1=2 1=2 1 1=2 .B f.A/B / D lim B f AEA , 1 B !1 n n 1=2 1 1=2 1=2 1=2 lim f B AEA , 1 B D.f.B AB //. n!1 n

Lemma A.2.4. Let A, B 2 M be positive operators. We have

1=2 1=2 1=2 1=2 B .A 1/CB .B AB 1/C provided that B 1 and .A/ f0g[.1, 1/.

Proof. Let t>.supp.A//. There exists >0suchthat.t , A/ D 0. It follows that .t, B1=2AB1=2/ ., B/ .t , A/ D 0.

If t<.supp.A//,then.t, A/ 1 and, therefore

1 1=2 1=2 1=2 1=2 1 .t, A/ kB k1 .t, B AB / .t, B AB /.

It follows that .B1=2AB1=2/ f0g[.1, 1/.(A.3)

Similarly, .B1=2supp.A/B1=2/ f0g[.1, 1/.(A.4)

We have

ker.B1=2AB1=2/ Df 2 H : B1=2AB1=2 D 0gDB1=2ker.A/.

Similarly, ker.B1=2supp.A/B1=2/ D B1=2ker.A/.

Therefore, supp.B1=2AB1=2/ D supp.B1=2supp.A/B1=2/.(A.5)

It follows from (A.4) and (A.5) that

supp.B1=2AB1=2/ B1=2supp.A/B1=2.(A.6)

It follows from (A.3) that

1=2 1=2 1=2 1=2 1=2 1=2 .B AB 1/C D B AB supp.B AB /.(A.7) Section A.2 Operator Inequalities 425

On the other hand, we have

1=2 1=2 1=2 1=2 1=2 1=2 B .A 1/CB D B AB B supp.A/B .(A.8)

The assertion follows from (A.6), (A.7) and (A.8).

Lemma A.2.5. Let A, B 2 M be positive operators. We have

1=2 1=2 1=2 1=2 .B .A 1/CB / ..B AB 1/C/ provided that B 1.

Proof. Let C D AEA.1, 1/.WehaveA C and, therefore,

B1=2AB1=2 1 B1=2CB1=2 1.

It follows from Lemma A.2.1 that

1=2 1=2 1=2 1=2 ..B AB 1/C/ ..B CB 1/C/.

It follows from Lemma A.2.4 that

1=2 1=2 1=2 1=2 1=2 1=2 .B CB 1/C B .C 1/CB D B .A 1/CB .

The assertion follows immediately.

Lemma A.2.6. Let A, B 2 M be positive operators. For every convex continuous increasing function f , we have

.B1=2f.A/B1=2 / .f.B1=2AB1=2// provided that B 1.

Proof. First, consider the case when supp.A/ is ﬁnite. We have

.supp.B1=2AB1=2//, .supp.B1=2f.A/B1=2// .supp.A//.

On the interval Œ0, kAk1 kBk1, the function f admits a uniform approximation with piecewise-linear convex increasing continuous functions. Hence, for every >0, there exists Xn g : t ! ˛k.t ˇk/C, kD1 such that ˛k, ˇk 0, 1 k n,and

f.t/ g.t/ f.t/C , 80 t kAk1 kBk1. 426 Appendix A Operator Results

By Lemma A.2.5, we have

.B1=2g.A/B1=2/ .g.B1=2AB1=2//.

It follows that

.B1=2f.A/B1=2 / .f.B1=2AB1=2// C .supp.A//.

Since >0 is arbitrarily small, the assertion is proved for ﬁnitely supported operator A. In the general case, we have 1=2 1=2 1=2 1 1=2 .B f.A/B / D lim B f AEA , 1 B !1 n n 1=2 1 1=2 1=2 1=2 lim f B AEA , 1 B D.f.B AB //. n!1 n

The following theorem is an immediate corollary of Lemma A.2.3 and Lemma A.2.6.

Theorem A.2.7. Let A, B 2 M be positive operators and let f be a convex continuous function such that f.0/ D 0. We have (a) .B1=2f.A/B1=2/ .f.B1=2AB1=2// if B 1.

(b) .B1=2f.A/B1=2/ .f.B1=2AB1=2// if B 1.

Lemma A.2.8. Let A, C 2 M be positive operators. We have (a) .1 C A/1 C.1 C CAC/1C if C 1.

(b) .1 C A/1 C.1 C CAC/1C if C 1.

Proof. We prove only the ﬁrst assertion. Let Cn :D maxfC ,1=ng, n 1. It follows that

1 C CnACn Cn.1 C A/Cn.

Hence, 1 1 1 1 .1 C CnACn/ Cn .1 C A/ Cn .

It follows that

1 1 1 .1 C A/ Cn.1 C CnACn/ Cn ! C.1 C CAC/ C .

If f is a power function, one can prove a signiﬁcantly stronger variant of Theo- rem A.2.7. Section A.2 Operator Inequalities 427

Lemma A.2.9. Let A, B 2 M be positive operators and let s>0. We have (a) .B1=2AB1=2/1Cs B1=2A1CsB1=2 if 0 B 1.

(b) .B1=2AB1=2/1Cs B1=2A1CsB1=2 if B 1.

Proof. For brevity, we set C :D B1=2. We use the following formula for the power of a positive operator D, Z sin.s/ 1 Ds D t sD.1 C tD/1dt. 0 It follows that Z sin.s/ 1 CA1CsC D t sCA.1 C tA/1ACdt. 0 and Z sin.s/ 1 .CAC /1Cs D t sCAC.1 C tCAC/1CACdt. 0 It follows that .CA1CsC .CAC /1Cs/ D sin.s/ Z 1 D t sCA..1 C tA/1 C.1 C tCAC/1C/ACdt. 0 The assertion now follows from Lemma A.2.8.

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Index

Amplitude 319 Classical Differential geometry see Riemannian Basis (of a Hilbert space) Geometry Eigen see Eigenbasis Pseudo-differential operator see Orthonormal 16, 26 Pseudo-differential operator, Bicommutant 39 Classical Theorem 39 Classical limit 385, 411, 413, 419 Bounded operator 15 Closed commutator subspace 154, 162 Absolute value 19 Closed Riemannian manifold 332, 363, Adjoint 16 376, see also Riemannian ge- Hermitian see Bounded operator, Self- ometry adjoint Isometry Group 396, 418 Normal 19, 41 Commutant 38 Positive 16, 39 Commutator subspace 154, 155, 182, Self-adjoint 16 190, 304, 380 Unitary 26 Closed see Closed commutator subspace C -algebra 39 History 192 Maximal such that the noncommuta- Of the weak-l1 ideal 191 tive integral is a hypertrace 393 Compact operator 17 Calculus -compact 53, 64 Connes’ quantum 303,306, 386, 395, Decomposition into normal and quasi- 417 nilpotent 22, 155,173, 189 Functional 19 Finite rank 16 Pseudo-differential 323,326, 333 Nilpotent 20, 173 Calkin Normal 19, 22, 155, 162, 173,188 Algebra 35 Quasi-nilpotent 20, 168, 173, 188 Correspondence see Calkin correspon- Compactly based operator 326, 350 dence Compactly supported operator 326, 350 Function space 53 Continuous trace see Trace, Continuous Operator space 53, 56 Correspondence Sequence space 23, 53, 153 Calkin see Calkin correspondence Calkin correspondence 22, 23, 27, 53, 54, Principle in quantum mechanics 413 57, 71, 75, 79, 117, 152 Cosphere bundle see Riemannian geome- Caratheodory theorem 82 try Cesàro operator Continuous 120, 278,281 Decomposition Discrete 154, 184 Jordan see Jordan decomposition 446 Index

Of a compact operator into normal and Invariant under isospectral deforma- quasi-nilpotent parts see Com- tion 398 pact operator, Decomposition Lidskii formula of 221, 232, 242,314, into normal and quasi-nilpotent 360 Of a self-adjoint operator into positive Linearity of 30 and negative parts 42, 157 Link between discrete and continuous Of a symmetric functional into hermi- versions 205 tian parts 110 Link to fully symmetric function- Of a symmetric functional into normal als 78, 195, 203, 222, 312 and singular parts 108, 123 Link to the noncommutative Of an operator into real and imaginary residue 304, 360, 370,389 parts 42, 157 Non-normality of 45 Polar 48 On a fully symmetric operator Schmidt 49 ideal 78, 206, 208, 213 Yosida-Hewitt 68, 70, 72 On a Lorentz function space 200, 226 Dilation On a Lorentz ideal 198, 199, 221, 226, Operator 85, 197 236 Semigroup 197 On a Lorentz operator space 218 Distribution 318 Wodzicki construction of 195, 216 Spectral see Spectral, Distribution Wodzicki representation, theorem 216 function Tempered 318 Eigenbasis Dixmier Of the Hodge-Laplacian see Lapla- -Macaev ideal see Ideal, Dixmier- cian, Eigenbasis Macaev Ordered 340, 363, 384, 387, 413 Measurable see Measurable operator, Eigenprojection see Spectral, Projection Dixmier Eigenvalue 17 Letter to the Luminy conference see (Algebraic) multiplicity 18 Dixmier, on the origin of Geometric multiplicity 17 Dixmier traces Matrix results on 420 On the origin of Dixmier traces 217 Of a Hodge-Laplacian modulated oper- Dixmier trace 29, 77, 194,221, 276, 296, ator 367 313, 316, 390, 411 Of a Laplacian modulated opera- Calculation of using -function residue tor 355 formulas 222, 245,267, 313 Eigenvalue sequence 18, 77, 153, 173, Calculation of using eigenvalues see 182,232, 297, 312, 340, 355, Lidskii formula of 366 Calculation of using expectation val- Eigenvalue theorem 355, 367 ues 241, 307,340, 361, 371, Expectation operator 101, 117 389 Expectation values 307, 340, 358, 365, Calculation of using heat kernel for- 384,391, 413 mulas 222,245, 251, 313, 414 Exponential map 287, 293 Connes’ inﬂuence on 152, 218, 300 Extended limit 196, 222 Continuous version of 199 Applied to the noncommutative Discrete version of 195, 205, 214 residue 305, 360 Distribution formulas for 226, 233, At 0 219 236 C -invariant 281, 291 Index 447

Composed with the logarithmic Harmonic function 168 mean 235, 391, 413 Heat kernel functional 222, 244, 246, Dilation and power invariant 267, 413 276, 313 Dilation invariant 30, 197, 228,245, Generalised 245, 256,259 250, 384 Link to classical limit and Gibbs M -invariant 221, 235, 245, 272, 293 states 414 On sequences 32, 196, 316 Link to fully symmetric function- On unbounded measurable func- als 245,255 tions 206, 227 High temperature limit 412 On unbounded positive sequences 206 Hilbert space 15, 153, 194, 339, 382, 386, 412 Fekete lemma 92 Of square integrable functions 40, Figiel-Kalton theorem 108, 120, 152,154 303, 331, 363,372, 396, 405 Closed version 163 Hodge-Laplacian modulated see Modu- Fredholm module see Unbounded Fred- lated operator, Hodge Laplacian holm Module Hypertrace 384, 393 Fubini’s theorem, a special form 171 Ideal Fully symmetric L1,1 (weak-l1) 26, 187, 297, 303, 315, Function space 59, 130 359 Ideal of compact operators 194, 206 M1,1 26, 29, 203, 221, 235, 297, 303, Operator space 59, 142 311 Trace see Trace, Fully symmetric M1,1.M, / 222, 244,246, 272, 276 With Fatou norm 59, 133, 143, 146 Arithmetically mean closed 72 Fully symmetric functional 69, 107, 130, Banach see Symmetric ideal 142 Banach sum 63 Approximation of by Dixmier Dixmier-Macaev (compact opera- traces see Dixmier Trace, On a tors) see Ideal, M1,1 fully symmetric operator ideal Dixmier-Macaev (general) see Ideal, Distinction from symmetric func- M1,1.M, / tional 76, 109, 133 Geometrically stable 173, 180, 193 Existence of 108, 130, 142 Lorentz see Lorentz, Ideal Relatively normal 210, 211 Of a semiﬁnite von Neumann alge- Functional bra 55 Fully Symmetric see Fully symmetric Of ﬁnite rank operators 24 functional Of nuclear operators see Trace class Minkowski see Minkowski functional operators Normal 68 Of trace class operators see Trace Singular 67 class operators Symmetric see Symmetric functional Orlicz 186 Unitarily invariant see Trace Schatten(-von Neumann) 24, 61, 63, 185, 192 Geometric stability see Ideal, Geometri- Symmetric see Symmetric ideal cally stable Symmetrically normed see Symmetric Gibbs state 412, 417 ideal Two-sided of compact operators 22, Hahn-Banach theorem 123 56 Invariant version 197 Weak-lp 24, 185 448 Index

Integral Lemma 175 As a faithful normal semiﬁnite Of symmetric function or sequence trace 44 spaces to symmetric operator Noncommutative see Noncommuta- spaces 97 tive Integral Of symmetric functionals on function Of functions (recovery of the Lebesgue spaces to operator spaces 75, integral) 305, 329, 334, 338, 107,117, 142, 200 372, 379, 406 Of symmetric sequence spaces to sym- On a manifold 331, 376,398 metric function spaces 101 Isometry group Limit Of a closed manifold see Closed Rie- Classical see Classical limit mannian manifold, Isometry Extended see Extended limit group Linear extension 27 Isospectral deformation 385,397 Logarithmic mean 221, 235, 244, 273, Of a smooth function 398,408 383,413 Of the torus see Noncommutative Of expectation values 308, 383,391, Torus 413 Tracial invariance of 398, 408 Lorentz Jordan decomposition Function space 65, 72, 199, 226 Of operator 232 Ideal of compact operators 25, 66, Of symmetric functional 109 194,198, 232 Operator ideal 66 , 200, 221, 273 Karamata theorem 279 Operator space 65, 67, 200 For dilation invariant extended limits 266 Marcinkiewicz see Lorentz Köthe dual see Lorentz operator space Change in terminology 72 Kwapien theorem 122 Mean Laplace-Beltrami operator see Laplacian, Cesàro see Cesàro operator Hodge Difference between Cesàro and Loga- Laplacian rithmic 417 Eigenbasis of 306, 338, 356, 363, 406 Logarithmic see Logarithmic mean Eigenvalues of see Weyl, Law Measurable operator Hodge 306, 332, 363, 396 Dixmier (or in Connes’ sense) 223, Modulated see Modulated operator, 272,276, 299, 313, 340, 360, Laplacian 371,383, 389, 390 On Euclidean space 304, 318, 321, Formulas for 314, 414 325, 336, 349 Link between Dixmier and M - 223, On the torus 405 272,290 Lidskii formula 33, 77, 153, 183, 312, 316 Link between Dixmier and Taube- Of a Dixmier trace see Dixmier trace, rian 299, 314, 316 Lidskii formula M - 223,272, 287 Of the canonical trace (Lidskii’s For- q- 383, 391, 414 mula) 185 Stronger version 390 Lifting Tauberian 297, 314 Kalton’s of subharmonic func- Minkowski functional 90, 95 tions 168 Modulated function 345, 374 Index 449

Modulated operator 336, 339 Noncommutative .H , D/- 387 Lp ideals see Ideal, Schatten(-von Hodge-Laplacian 338, 363, 377, 385, Neumann) 397 Lp spaces 53, 60 Isospectral deformation of 397 Integration see Trace, Normal and see Laplacian 304, 336, 349, 360, 373 Trace, Semifinite Localised 367 Integration à la Connes 68, 70, see Pseudo-differential operator as an ex- also Noncommutative Integral ample of 351, 364 Measure theory see Von Neumman Set of as a bimodule for the pseudo- algebra, Semifinite differential operators of order Pseudo-differential operators see Cal- 0 364 culus, Connes’ quantum Mollifier 318 Noncommutative Geometry 303, 382 Moyal product 397, 418 Dimension in see Unbounded Fred- Multi-index 317 holm Module, Dimension of Examples in using singular traces 309 Von Neumann algebra 38 Noncommutative Integral 307, 383, 391, -finite 43, 49 414, 417 Atomic 40, 45, 58, 69, 79, 108, 118, As a hypertrace 393, 408 143, 153, 194, 246, 272 Extension of the Lebesgue integral see Atomless 40, 45, 52, 58, 69, 79, 108, Integral, Of functions 118, 143, 218, 240, 246, 272 Invariant of isospectral deforma- Bimodule of 55, 68 tion 398 Link to classical limits 308, 383, 413 Center of 39, 55 Noncommutative residue Commutative 39 Extension of (vector-valued) 304, Factor 39, 55, 68, 69, 153 306, 337, 352,365, 385, 396 Finite 43 In Noncommutative Geometry 307, Lattice of projections see Projection, 382, 389 Lattice Invariance under isospectral deforma- Maximal commutative 39 tion 385, 397 Of bounded operators on a separable Link to integration see Integral, Of Hilbert space 15, 38, 108, 153, functions 194 Link to singular traces 304, 337, 360, Of essentially bounded functions 40, 370 44, 63, 407 Spectral formula 306,338, 370, 385 Pre-dual 62 Wodzicki (scalar-valued) 305, 329, Semifinite 43, 68 334, 362, 369 Nigel Kalton Noncommutative Torus 405, 408 Origin of modulated operator 380 Trace on 408 Origin of the description of the com- Norm mutator subspace of the trace Lp 60 class operators 192 Dual 66 Origin of uniform submajoriza- Fatou 59 tion 104 Operator see Operator norm Non-measurable operator 314 Quasi 187, 315 Non q-measurable 392 Symmetric see Symmetric norm Pseudo-differential 306,362, 371 Uniform see Operator norm 450 Index

Operator Spectral see Spectral, Projection -measurable 46 Support 42 Afﬁliated 46 Projection valued measure 41 Bimodule see Von Neumann algebra, Pseudo-differential operator 303, 319 Bimodule of Adjoint 319 Bounded see Bounded operator Bounded extension of 325, 333 Cesàro see Cesàro operator Classical 304, 328, 333, 362, 375 Compact see Compact Operator Compact extension of 327, 333 Diagonal 19, 153, 184 Compactly based see Compactly Dilation see Dilation operator based operator Expectation see Expectation operator Compactly supported see Compactly Finite rank 16, 50 supported operator Ideal see Ideal Kernel 320 Imaginary part 42 On a manifold 332 Inequalities 422 Power of 333 Localised see Modulated operator, Regularity of 324, 327 Localised Shubin 327,351 Measurable see Measurable operator Symbol of see Symbol Modulated see Modulated operator Trace class 327 Normal see Compact operator, Nor- Trace on order 0 390 mal or Bounded operator, Nor- Pseudo-differential operator of order mal d 303, 329 Nuclear see Operator, Trace class As a Laplacian modulated opera- Pseudo-differential see Pseudo- tor 351, 364 differential operator Eigenvalues 355 Real part 42 Isospectral deformation of 397 Space see Symmetric operator space Non-uniqueness of traces on see Non- Tauberian see Measurable operator, Measurable operator, Pseudo- Tauberian differential Trace class 29 Unique trace on classical see Non- Operator norm 15, 38, 60 commutative residue, Wodz- Operator topology icki and see Trace Theorem, Measure 71 Connes’ Norm 38 Strong 38, 397 Quantum ergodicity 384, 413 Uniform see Operator topology, Norm Quasi-nilpotent see Compact operator, Weak 38 Quasi-nilpotent Weak 212 Radon-Nikodym Theorem see Trace, Peter-Weyl form 398 Duality Projection 39 Rapid decrease see Schwartz functions -ﬁnite 43, 64 Renormalization (multiplicative) 195 Abelian 41 Riemannian geometry 330, 396 Associated to isospectral deforma- Partition of unity 330, 367 tion 397 Riemannian metric 330, 354, 364, 368, Finite rank 16, 24 396 Lattice of 39 Riesz theorem, on subharmonic func- Rank one 16, 41 tions 171 Index 451

Schur-Horn problem, Kaftal and Weiss Spectral contribution to resolution Distribution function 48, 226, 421 of 241 Geometry 363 Schwartz functions 318, 356 Measure 42, 48 Semifinite Projection 42 Trace see Trace, Semifinite Theorem 19, 41 Von Neumann algebra see von Neu- Spectral triple see Unbounded Fredholm mann algebra, Semifinite Module Sequence space Spectrum 17, 233 l 24, 61, 185 Sphere (d 1) 328 p Square integrable functions see Hilbert Lorentz 24, 226 space, of square integrable Orlicz 186 functions Sargent 25, 311 State, on a von Neumann algebra 196 Separable part 164 Stone–Cechˇ compactification 195, 214 Weak-lp 24, 185,187 Subharmonic Singular state Function 168 L On .H / 384 Mapping 168 On l1 or L1 see Extended Limit Submajorization Singular trace 33, 70, 304, 312, 316, 411 Hardy-Littlewood(-Polya) 58, 69, 83, Invariant of isospectral deforma- 312 tion 398 In the finite-dimensional setting 80 Link to classical limits 308, 385, 411 Uniform 79, 88 Link to the Lebesgue integral 305, Symbol 338, 372 Asymptotic expansion of 323 Link to the noncommutative Compactly supported in the first vari- residue 304, 337, 360, 370, able 326,328, 350 389 Of a Hodge-Laplacian modulated oper- On the weak-l1 ideal 191 ator 367 Uniqueness of 312 Of a Laplacian modulated opera- Singular value tor 305,336, 349, 350, 353, Function 48, 71, 79, 107 355, 360 Sequence 19, 23, 49, 77, 153 Of a Pseudo-differential operator 320 Principal 321,329, 333, 353, 367 Singular values (of a compact opera- Symmetric tor) 19, 23, see also Singular Function space 56, 72, 76, 97, 117 value sequence Norm 28, 56,97 Product 21 Sequence space 56, 76, 101, 117, 153, Sum 21, see also Submajorization, 162, 180 Hardy-Littlewood(-Polya) Symmetric functional 69, 107,123 Slow oscillation at infinity see Taube- Existence of 108, 120, 123, 142 rian, Theorem of Kwee Hermitian 109 Smooth functions 318, 330, 375 In the Figiel-Kalton sense 152 Deformation of see Isospectral defor- Lattice of 116, 146 mation, of a smooth function Lifting of from a function space to an Smooth manifold see Riemannian geom- operator space see Lifting, Of etry symmetric functionals from Sobolev space 324, 333 function spaces to operator Connes-Moscovici approach 386 spaces 452 Index

Link to traces 69, 77, 107, 153 Fully symmetric 69, 194, 312 Positive part 109 Hilbert space see Trace, Canonical Singular 70, 108, 123 Hyper see Hypertrace Symmetric ideal 56 Matrix 22 Fully symmetric see Fully symmetric, Normal 43, 61, 68, 70, 217 Ideal Normalised 311 Of compact operators 28, 56, 70, 71, On L1,1 190 77, 153, 173, 194 On a pseudo-differential operator of Which is not fully or strongly symmet- order d see Noncommutative ric 72 residue Symmetric operator space 56, 63, 68, 97, On a pseudo-differential operator of 117, 142 order 0 390 Correspondence with a symmetric Semiﬁnite 42, 68 function space see Lifting, of Singular see Singular trace symmetric function spaces to Spectrality of see Lidskii formula symmetric operator spaces Wodzicki construction of 195 Fully symmetric see Fully symmetric, Trace class operators 28, 29, 62, 185, Operator space 311,384 Lorentz see Lorentz, Operator space Pseudo-differential operators as see Separable part 67, 70, 311 Pseudo-differential operator, Strongly symmetric 59, 66 Trace class Which is an ideal see Symmetric ideal Trace Theorem 305, 337, 360 Symmetrically normed ideal see Sym- Connes’ 304, 362, 372, 380 metric ideal, Of compact opera- In Noncommutative Geometry 307, tors 389 On a manifold 370 Tangent bundle see Riemannian geometry Unbounded Fredholm module 382, 386, Tauberian 391,394, 413 Operator see Measurable operator, Dimension of 307, 327, 386, 417 Tauberian Of a closed Riemannian manifold 396 Theorem of Kwee 384, 392 Of the torus 405 Variant of Hardy’s theorem, Unitary action (strongly continuous) lemma 278 Of the isometry group 396 Topology see Operator topology Unitary operator see Bounded operator, Torus 396, 405 Unitary Noncommutative see Noncommuta- tive Torus Weak-l1 ideal see Ideal, L1,1 Trace 26, 68, 316, 340 Weyl As a symmetric functional 69, 70, 77, Asymptotic condition (noncommuta- 153 tive geometry) 308, 383, 387, Canonical 29, 44, 185 392,413 Construction of 27, 36, 69, 194, 216 Law 357, 363, 406 Continuous 28, 68, 70, 183, 311 Lemmas 21 Dixmier see Dixmier trace Relations (noncommutative torus) 408 Duality 62, 67, 73 Existence of 109, 120, 184,194 -function 244 Faithful 43 -function residue 222, 244, 263, 276, Finite 43 314