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Dixmier Traces 194 6.1 Introduction 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 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper 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 discov- ered with Nigel, based on previous work about symmetric spaces, symmetric func- tionals 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, noncom- mutative 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 publica- tion 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 à 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 <A (=A) real (imaginary) part of the operator A A˚n direct sum of n-copies of the operator A Dom.A/ domain of the operator A ker.A/ kernel of the operator A .A/ spectrum of the operator A .A/ an eigenvalue sequence of the operator A .n, A/ n-th element of an eigenvalue sequence of the operator A .A/ singular value sequence (or function) of the operator A .n, A/ n-th element of the singular value sequence of the operator A .t, A/ value at t 2 Œ0, 1 of the singular value function of the operator A J Calkin 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 func- tions 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 semifi- nite 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 :0Ä 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 .
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