Dixmier Traces and Some Applications in Non-Commutative Geometry

Russian Math. Surveys 61:6 1039–1099 c 2006 RAS(DoM) and LMS Uspekhi Mat. Nauk 61:6 45–110 DOI 10.1070/RM2006v061n06ABEH004369

Dixmier traces and some applications in non-commutative geometry

A. L. Carey and F. A. Sukochev

Abstract. This is a discussion of recent progress in the theory of singular traces on ideals of compact operators, with emphasis on Dixmier traces and their applications in non-commutative geometry. The starting point is the book Non-commutative geometry by Alain Connes, which contains several open problems and motivations for their solutions. A distinctive feature of the exposition is a treatment of operator ideals in general semiﬁnite von Neumann algebras. Although many of the results presented here have already appeared in the literature, new and improved proofs are given in some cases. The reader is referred to the table of contents below for an overview of the topics considered.

Contents

1. Introduction 1040 2. Preliminaries: spaces and functionals 1043 2.1. Marcinkiewicz function spaces and sequence spaces 1044 2.2. Singular symmetric functionals on Marcinkiewicz spaces 1044 2.3. Symmetric operator spaces and functionals 1045 3. General facts about symmetric functionals 1047 4. Preliminaries on dilation- and translation-invariant states 1049 5. Concrete constructions of singular symmetric functionals 1054 5.1. Dixmier traces 1054 5.2. Connes–Dixmier traces 1056 5.3. Rearrangement-invariant functionals and singular traces 1058 6. Class of measurable elements 1058 7. Norming properties of Dixmier and Connes–Dixmier functionals 1061 8. Fredholm modules and spectral triples 1063 8.1. Notation and definitions 1063 8.2. Bounded versus unbounded modules 1064 8.3. More on semifinite spectral triples 1064 8.4. Summability and dimension 1065 9. Spectral flow 1066 9.1. Spectral flow formulae 1067 9.2. Relation to cyclic cohomology 1067

AMS 2000 Mathematics Subject Classiﬁcation. Primary: 46L10, 58B34, 58J30, 58J42, 47L20. 1040 A. L. Carey and F. A. Sukochev

10. The Dixmier trace and residues of the zeta function 1069 10.1. Preliminaries 1069 10.2. The zeta function and the Dixmier trace 1074 11. The heat semigroup formula 1077 12. The case p > 1 1079 13. Generalized Toeplitz operators and their index 1081 14. Non-smooth foliations and pseudodiﬀerential operators 1082 15. The algebra of almost periodic functions 1084 15.1. Almost periodic pseudodiﬀerential operators 1084 15.2. Almost periodic spectral triple 1088 16. Lesch’s index theorem 1088 17. The Hochschild class of the Chern character 1091 18. Lidskii-type formula for Dixmier traces 1093 Bibliography 1095

1. Introduction

The Dixmier trace τω arose from the problem of whether the algebra B(H ) of all bounded linear operators on a Hilbert space H had a unique non-trivial trace. Dixmier resolved this question in the negative in 1966 in a note in Comptes Rendus [1]. To construct the trace he used an invariant mean ω on the soluble ‘ax + b group’. His trace vanishes on the ideal of trace-class operators and hence is completely disjoint from the usual trace. It is also non-normal. Applications of the Dixmier trace to classical geometry are facilitated by a remarkable result [2] relating the trace to the Wodzicki residue for pseudodifferential operators on a closed manifold [3]. The latter was introduced independently by Adler [4] and Manin [5] in the one-dimensional case, and developed further by Wodzicki [3] and Guillemin [6] in higher dimensions. The intriguing and most useful property of the Wodzicki residue apart from its computability in examples is the fact that it makes sense for pseudodifferential operators of arbitrary order. Moreover, it is the unique trace on the pseudodifferential operators which extends the Dixmier trace on operators of order not exceeding minus the dimension of the underlying manifold [2]. In non-commutative geometry and its applications both the Dixmier trace and the Wodzicki residue play important roles [7], [8]. In particular, the Dixmier trace τω is an appropriate non-commutative analogue of integration on a compact n-dimensional Riemannian spin manifold M; more precisely, Z −n 1 [n/2] τω(f(x)|D| ) = n 2 Ωn−1 f(x) dvol, (1) (2π) n M where f is a smooth function on M, D is the Dirac operator on M,Ωn−1 is the volume of the (n − 1)-dimensional unit sphere, and dvol is the Riemannian volume form. The usefulness of the Dixmier trace was extended by results of Connes [7], who connected it with residues of zeta functions. Some important applications which we do not have the space to include are to the theory of gravitation, classical field theory, and particle physics. They are well covered in the book [8] and stem from the relationship between the Wodzicki residue Dixmier traces and some applications 1041 of powers of the Dirac operator and the Einstein–Hilbert action [2]. Likewise, we do not try to cover other material in the books [7] and [8], which describe several interesting physical applications of the Dixmier trace. We refer the reader to the survey [9] and the extensive literature on applications of non-commutative geometry to the ‘standard model’ of particle physics, where, starting with an elegant expression for a non-commutative action principle expressed in terms of the Dixmier trace, Connes and Lott [10] and Connes [11] show how to derive from it the Euclidean version of the action for the standard model. Analogously, one can also give a non-commutative formulation of the Hamiltonian version of classical field theory, again by using the Dixmier trace [12]. Some background to these developments is provided by [13], [14]. These applications to physics can be traced back to the research announce- ment [15], where a series of fundamental results are described (some of these were later collected in [7]). After calculating the Dixmier trace for pseudodifferential operators, Connes obtained the Yang–Mills and Polyakov actions from an action functional involving the Dixmier trace. The theorem on a residue formula for the Hochschild class of the Chern character, which we will describe in § 17, was also announced there. The next step appears in [16], where Connes introduced the axioms of non-commutative spin geometry for a non-commutative algebra A . The very first axiom uses the Dixmier trace to introduce a non-commutative integration theory on A which is completely natural in view of (1). This point of view resurfaces in our discussion of Lesch’s index theorem in § 16, although we will not digress further to introduce the other axioms of Connes except to mention in § 17 the role of the Hochschild class. While these applications to physical theory form a motivational background, they are not the focus of this article. Our aim is to give a unified and coherent account of some recent functional analytic advances in the theory of Dixmier traces. In addition to surveying these new results we also offer in some cases new proofs. Part of our motivation is to extend and clarify questions raised by [7], Ch. IV. Specifically, we characterize the class of measurable operators defined in [7], explain the role of the Cesàromean in Connes’ version of Dixmier traces (called Connes– Dixmier traces here), and give a complete analysis of zeta function formulae for the Dixmier trace. Most importantly, however, our whole treatment is within the framework of ‘semifinite spectral triples’, which we explain in § 8. This notion arises when one extends the theory in [7], which deals with subalgebras of the bounded operators on a separable Hilbert space that are equipped with the standard trace, to the situation where the von Neumann algebra of bounded operators is replaced by a general semifinite von Neumann algebra equipped with some faithful normal semifinite trace [17]. As a result of the extra effort needed to handle this level of generality it is possible to find improvements even in the standard type I theory developed in [7]. A number of authors have contributed to the development of this approach and of new applications of Dixmier traces and more general singular traces [18]–[39]. Our exposition is organized around three issues. The first issue dominates the early part of the article (culminating in § 6) and gives a characterization of measurable operators in the sense of [7], § IV.2. We approach the topic from the study of the general theory of singular symmetric functionals (this and its preliminaries 1042 A. L. Carey and F. A. Sukochev occupy §§ 2–5). Our exposition is based on the approach articulated in [31], [29], which considers such traces as a special class of continuous linear functionals on the corresponding operator ideals. Many features of the theory may be well under- stood, even in the most trivial situation when the von Neumann algebras in question are commutative. In this situation the theory of Dixmier traces roughly corre- sponds to the theory of symmetric functionals on rearrangement-invariant function spaces [40], [41], [31] and allows an alternative treatment based on the methods drawn from real analysis. In preparation for the rest of the paper we then introduce the notions of semifinite spectral triple, type II spectral flow and some notation for cyclic cohomology. This leads us to the second issue, occupying §§ 10 and 12, where we describe the expression of the Dixmier trace in terms of residues of zeta functions. The key observation, made in § 4, that enables us to prove a significant generalization of Proposition IV.2.4 in [7] and also of the measurability theorem in § 6, is the existence of two kinds of Dixmier trace. One is associated with the multiplicative group of the positive reals and its invariant mean, and the second, which is associated naturally with the zeta function, arises from the invariant mean of the additive group of the reals. The relationship between the two captures the formula for the Dixmier trace in terms of the zeta function. These results use the language of spectral triples and lead naturally to the third issue, namely, some applications, which begin in § 11. We give a proof of (an extension of) the heat semigroup formula of [7] (p. 563) for the Dixmier trace. We then give a formula for the index of generalized Toeplitz operators (§ 13), describe a special case of the Wodzicki residue formula of Benameur and Fack [22] for pseudodifferential operators along the leaves of a foliation, and discuss in § 15 a similar formula for pseudodifferential operators with almost periodic coefficients. The formula of § 13 has as a consequence the index theorem of Lesch for Toeplitz operators with non-commutative symbol, and this is described in § 16. An extension of Theo- rem IV.2.8 of [7] on a residue formula for the Hochschild class of the Chern character of Fredholm modules is given in § 17 (see also [22]). Finally, following [21], we show how to extend Lidskii’s formula to Dixmier traces in the von Neumann setting in § 18. In a series of corollaries we explain its relevance to the question of measurability of certain classical operators. In [7] measurability results are proved for positive operators, except for the case of pseudodifferential operators. The approach of [21] allows one to address the problem of removing the positivity assumption. We present below a short list of symbols and terminology used in this paper, with indication of the place where these symbols and notations are introduced: • symmetric and rearrangement-invariant functionals and rearrangement- invariant spaces E(J) (Definitions 2.1 and 2.3); • Marcinkiewicz spaces, M(ψ), L (1,∞), L (p,∞) (§ 2.1); • generalized singular-value function µ( · )(x)(§ 2.3); • semifinite von Neumann algebra N , faithful normal semifinite trace τ on N , the set Nf of τ-measurable operators, the fully symmetric operator space E(N , τ) associated with (N , τ) and a Banach function space E (§ 2.3); • operator Marcinkiewicz spaces M(ψ)(N , τ)(§ 2.3); ∗ • symmetric functionals on a fully symmetric space E = E(0, ∞): Esym (§ 3); Dixmier traces and some applications 1043

∗ • the sets BL(R), D(R+), BL(R+) of states (§ 4);

• Dixmier traces τω (Definitions 5.1, 5.4), the traces FL,FL (§ 5.1); ∗ • Connes–Dixmier traces τω with ω ∈ CD(R+)(§ 5.2); • measurable operators (Definitions 6.1 and 6.2); • spectral triples (A , H , D), grading operator Γ, I -summability (see Defini- tion 8.1); • semifinite spectral triples of class QCk (Definition 8.6); • spectral dimension of (A , H , D) (Definition 8.7); ∗ • spectral flow sf({Ft}), sf(D, uDu )(§ 9); • (b, B)-cochain, (b, B)-cocycle, (bT ,BT )-chain, (bT ,BT )-cycle (§ 9);

• zeta functions ζ(s), ζA(s)(§ 10.2); n ∗ n • Bohr compactiﬁcation RB, C -algebra A P (R )(§ 15.1). The authors would like to thank N. A. Azamov and A. A. Sedaev for comments, assistance, and critical remarks.

2. Preliminaries: spaces and functionals

We consider a Banach space (E, k · kE), of real-valued Lebesgue-measurable functions x (with identiﬁcation almost everywhere) on the interval J = [0, ∞) or else on J = N. Let x∗ denote the non-increasing right-continuous rearrangement of |x|, given by ∗ x (t) = inf{s > 0 : λ({|x| > s}) 6 t}, t > 0, where λ denotes Lebesgue measure. Then E will be called rearrangement invariant if: (i) E is an ideal lattice, that is, if y ∈ E and x is any measurable function on J with 0 6 |x| 6 |y|, then x ∈ E and kxkE 6 kykE; (ii) if y ∈ E and if x is any measurable function on J with x∗ = y∗, then x ∈ E and kxkE = kykE. For J = N it is convenient to identify x∗ with the non-increasing rearrange- ∞ ment of the sequence |x| = {|xn|}n=1. See the monographs [42]–[44] for basic properties of rearrangement-invariant spaces. We note that for any rearrangement- invariant space E = E(J) the following continuous embeddings hold:

L1 ∩ L∞(J) ⊆ E ⊆ L1 + L∞(J).

A rearrangement-invariant Banach space E is said to be fully symmetric if it has the additional property that if y ∈ E and L1 +L∞(J) 3 x ≺≺ y, then x ∈ E and kxkE 6 kykE. Here, x ≺≺ y denotes submajorization in the Hardy–Littlewood–P´olya sense:

Z t Z t ∗ ∗ x (s) ds 6 y (s) ds ∀ t > 0. 0 0

A classical example of non-separable fully symmetric function and sequence spaces E(J) is given by Marcinkiewicz spaces. 1044 A. L. Carey and F. A. Sukochev

2.1. Marcinkiewicz function spaces and sequence spaces. Let Ω be the set of concave functions ψ : [0, ∞) → [0, ∞) with limt→0+ ψ(t) = 0 and limt→∞ ψ(t) = ∞. Important functions belonging to Ω include t, log(1 + t), tα and (log(1 + t))α for 0 < α < 1. Let ψ ∈ Ω. We deﬁne the weighted mean function

1 Z t a(x, t) = x∗(s) ds, t > 0, ψ(t) 0 and denote by M(ψ) the Marcinkiewicz space of measurable functions x on [0, ∞) such that kxkM(ψ) := sup a(x, t) = ka(x, · )k∞ < ∞. (2) t>0

The deﬁnition of the Marcinkiewicz sequence space (m(ψ), kxkm(ψ)) is similar:

N 1 X m(ψ) = x = {x }∞ : kxk := sup x∗ < ∞ . n n=1 m(ψ) ψ(N) n N>1 n=1

1 The norm closure of M(ψ) ∩ L ([0, ∞)) (respectively, of `1 = `1(N)) in M(ψ) (respectively, in m(ψ)) is denoted by M1(ψ) (respectively, m1(ψ)). For any ψ ∈ Ω we have M1(ψ) 6= M(ψ). The Banach spaces (M(ψ), k · kM(ψ)), (m(ψ), k · km(ψ)), (M1(ψ), k · kM(ψ)), and (m1(ψ), k · km(ψ)) are examples of fully symmetric spaces [43], [42]. Let M+(ψ) (respectively, m+(ψ)) denote the set of positive functions in M(ψ) (respectively, m(ψ)). For any x ∈ M(ψ) we write x = x+ − x−, where x+ := xχ{t:x(t)>0} and x− := x − x+. The spaces

(1,∞) (p,∞) 1− 1 L := M(log(1 + t)) ∩ L∞ and L := M(t p ) ∩ L∞, 1 < p < ∞, play very important roles in what follows. We remark that these spaces are still Marcinkiewicz spaces. Indeed, L (1,∞) (respectively, L (p,∞), p > 1) can be identi- ﬁed with the space M(ψ1) (respectively, M(ψp), p > 1), where ( ( t · log 2, 0 6 t 6 1, t, 0 6 t 6 1, ψ1(t) = and ψp(t) = 1− 1 p > 1. log(1 + t), 1 6 t < ∞, t p , 1 6 t < ∞,

The (Marcinkiewicz) norm given by the formula (2) on the space L (p,∞) is denoted by k · k(p,∞), 1 6 p < ∞. 2.2. Singular symmetric functionals on Marcinkiewicz spaces. Deﬁnition 2.1. A positive functional f ∈ M(ψ)∗ is said to be symmetric (respectively, rearrangement invariant) if f(x) 6 f(y) for all x, y ∈ M+(ψ) such that x ≺≺ y (respectively, x∗ = y∗). Such a functional is said to be supported at inﬁnity (or sin- ∗ gular) if f(|x|) = 0 for all x ∈ M1(ψ) (equivalently, f(x χ[0,s]) = 0 for all x ∈ M(ψ) and for the characteristic function χ[0,s] of the interval [0, s] for all s > 0). The following theorem completely characterizes Marcinkiewicz spaces admitting non-trivial symmetric functionals. Dixmier traces and some applications 1045

Theorem 2.2 ([31], [40], [41]). If ψ ∈ Ω, then: (i) a non-zero symmetric functional on M(ψ)(respectively, m(ψ)) supported at inﬁnity exists if and only if

ψ(2t) lim inf = 1, (3) t→∞ ψ(t)

(ii) a non-zero symmetric functional on M(ψ) supported at zero exists if and only if ψ(2t) lim inf = 1. (4) t↓0 ψ(t)

(1,∞) ∗ (p,∞) ∗ Thus, for example, (L )sym,∞ 6= {0}, whereas (L )sym,∞ = {0}, for all 1 < p < ∞. The conditions (3) and (4) admit the following geometric interpreta- tion. We denote by N(ψ) the norm closure in M(ψ) of the (order) ideal

0 ∗ 0 N(ψ) := {f ∈ M(ψ): f ( · ) 6 kψ ( · /k) for some k ∈ N}. Clearly, N(ψ) is a Banach function space (a subspace of M(ψ)) and is rearrangement invariant. Assuming (for simplicity) that ψ is linear in a neighbourhood of 0, we get that N(ψ) is fully symmetric (and thus coincides with M(ψ)) if and ∗ only if (3) fails. In other words, M(ψ)sym,∞ 6= {0} if and only if N(ψ) 6= M(ψ). For this and other geometric interpretations of conditions (3) and (4) we refer the reader to [45], [42] (II.5.7), and [46]. For various constructions of singular symmetric functionals on M(ψ) (and, more generally, on fully symmetric spaces and their non-commutative counterparts) we refer to [31], [40], [41]. Constructions relevant to our main topic will be reviewed below, in § 5. Our focus on symmetric functionals supported at infinity is due to the numerous applications of their analogues in non-commutative geometry. Non-commutative analogues of symmetric functionals supported at zero can be thought of as Dixmier traces associated with von Neumann algebras of type II1 and have not found any applications to date. 2.3. Symmetric operator spaces and functionals. Here we extend the ideas of the preceding subsections to the setting of (non-commutative) spaces of measurable operators. We denote by N a semifinite von Neumann algebra on the Hilbert space H , with a fixed faithful and normal semifinite trace τ. We shall be mainly concerned with the case when τ(1) = ∞, where 1 is the identity in N . A linear operator x: dom(x) → H with domain dom(x) ⊆ H is said to be affiliated with N if ux = xu for all unitary elements u in the commutant N 0 of N . A closed and densely defined operator x affiliated with N is said to be τ-measurable if for any ε > 0 there exists an orthogonal projection p ∈ N such that p(H ) ⊆ dom(x) and τ(1 − p) < ε. The set of all τ-measurable operators is denoted by Nf. We next recall the notion of generalized singular-value function [47], [48]. Given a self-adjoint operator x acting in H , we denote by ex( · ) the spectral measure of x. Assume now that x is τ-measurable. Then e|x|(B) ∈ N for all Borel sets |x| B ⊆ R, and there exists an s > 0 such that τ(e (s, ∞)) < ∞. For t > 0 we define

|x| µt(x) = inf{s > 0 : τ(e (s, ∞)) 6 t}. 1046 A. L. Carey and F. A. Sukochev

The function µ(x): [0, ∞) → [0, ∞] is called the generalized singular-value function (or decreasing rearrangement) of x; note that µt(x) < ∞ for all t > 0. For the basic properties of this singular-value function we refer the reader to [47]. If we regard N = L∞([0, ∞), m), where m denotes Lebesgue measure on [0, ∞), as an Abelian von Neumann algebra acting via multiplication on the Hilbert space H = L2([0, ∞), m), with the trace τ given by integration with respect to m, then it is easy to see that the set of all τ-measurable operators affiliated with N consists of all measurable functions on [0, ∞) which are bounded except on a set of finite measure, and that the generalized singular-value function µ(f) is precisely the decreasing rearrangement f ∗. If N = L (H ) (respectively, `∞(N)) and τ is the standard trace Tr (respectively, the counting measure on N), then it is not difficult to see that Nf= N . In this case, x ∈ N is compact if and only if limt→∞ µt(x) = 0; moreover,

µn(x) = µt(x), t ∈ [n, n + 1), n = 0, 1, 2,...,

∞ and {µn(x)}n=0 is just the sequence of eigenvalues of |x| in non-increasing order and counted according to multiplicity. For a semiﬁnite von Neumann algebra (N , τ) and a fully symmetric Banach function space (E, k · kE) on ([0, ∞), m), we deﬁne the corresponding non-commutative space E(N , τ) by setting

E(N , τ) = {x ∈ Nf: µ(x) ∈ E}.

Equipped with the norm kxkE(N ,τ) := kµ(x)kE, the space (E(N , τ), k · kE(N ,τ)) is a Banach space and is called the (non-commutative) fully symmetric operator space associated with (N , τ) and corresponding to (E, k · kE). If N = `∞(N), then the space E(N , τ) is simply the (fully) symmetric sequence space È, which may be viewed as the linear span in E of the vectors en = χ[n−1,n), n > 1 (see, for example, [43]). In the case (N , τ) = (L (H ), Tr) we denote E(N , τ) simply by E(H ). Note that the latter space coincides with the (symmetrically-normed) ideal of compact operators on H associated with the (symmetric) sequence space È (see, for example, [49]). We shall be mostly concerned with fully symmetric operator spaces E(N , τ) (p,∞) and E(H ) when E = M(ψ), in particular, when E = L , 1 6 p < ∞. The spaces M(ψ)(N , τ) will be called operator Marcinkiewicz spaces. For brevity we shall sometimes omit the symbols (N , τ) and H from the notation if no confusion should result. Further references to the theory of fully symmetric operator spaces can be found in [31], [50], [51]. Definition 2.3. A linear functional ϕ ∈ E(N , τ)∗ is said to be symmetric (respectively, rearrangement invariant) if ϕ is positive (that is, ϕ(x) > 0 whenever 0 6 x ∈ E(N , τ)) and ϕ(x) 6 ϕ(y) whenever µ(x) ≺≺ µ(y) (respectively, ϕ(x) = ϕ(y) whenever x, y > 0 and µ(x) = µ(y)). For a given x ∈ Nf, the set Ω(x) = {y ∈ Nf: y ≺≺ x} is called the orbit of the operator x. If x ∈ L1(N , τ) + N , then Ω(x) is conveniently described in terms of Dixmier traces and some applications 1047 absolute contractions. Let Σ be the set of all linear operators T : L1(N , τ) + N → L1(N , τ) + N such that (a) T (a) ∈ L1(N , τ) (respectively, N ) if a ∈ L1(N , τ)

(respectively, N ), and (b) kT kL1(N ,τ)→L1(N ,τ) 6 1 and kT kN →N 6 1. It follows from [52] that y ≺≺ x with x ∈ L1(N , τ) + N and y ∈ Nfif and only if there exists a T ∈ Σ such that T (x) = y. Thus,

Ω(x) = {T x : T ∈ Σ}.

If E(N , τ) is a fully symmetric operator space, then Ω(x) ⊆ E(N , τ) for every x ∈ E(N , τ), and therefore a bounded positive linear functional ϕ on E(N , τ) is symmetric if and only if ϕ(|T x|) 6 ϕ(x) for all T ∈ Σ and 0 6 x ∈ E(N , τ). We now assume that α: N → N is a ∗-automorphism which is in addition trace preserving, that is, τ(α(a)) = τ(a) for all 0 6 a ∈ N . It is easy to see that such an automorphism extends (uniquely) to a ∗-automorphism αe: Nf→ Nf which is rearrangement preserving, that is, µ(αe(x)) = µ(x) for all x ∈ Nf. Thus, we can view rearrangement-invariant functionals on E(N , τ) as positive functionals which are invariant with respect to the action of the group of all trace preserving ∗-automorphisms of N , which is a subgroup of Σ. If N = L (H ), then every rearrangement-invariant functional on E(H ) is simply a trace (that is, a unitary-invariant positive functional on L (H )), which extends to a continuous linear functional on E(H ). Clearly, every symmetric functional is rearrangement invariant. However, a priori it is not clear (see [37]) whether there exists (for example on the ideal L (1,∞)(H )) a rearrangement-invariant singular functional which is not necessarily symmetric, or whether there exists a trace on L (1,∞)(H ) which does not coincide with a Dixmier trace (see § 5 below). Very recently the ﬁrst example of such a trace appeared in [35]. In fact, if

ψ(2t) lim = 1, (5) t→∞ ψ(t) then there exists on every operator Marcinkiewicz space M(ψ)(N , τ) a non-zero trace (or rearrangement-invariant functional) which vanishes on N(ψ)(N , τ). In particular, if ψ(t) = log(1 + t), then such a functional vanishes on every operator (1,∞) 0 6 x ∈ L (N , τ) with µt(x) = 1/t (or if N = B(H ), then on every compact (1,∞) operator 0 6 x ∈ L (H ) such that µn(x) = 1/n , n > 1). It is not clear yet whether rearrangement-invariant functionals which are not symmetric exist on every space M(ψ)(N , τ) with ψ satisfying condition (3).

3. General facts about symmetric functionals

Let E = E(0, ∞) be a fully symmetric space. By E+ we denote the set of all ∗ non-negative functions in E and by Esym the set of all symmetric functionals on E. A positive linear functional ϕ on E is said to be normal (or order continuous) if from fn ↓ 0 it follows that ϕ(fn) ↓ 0.

∗ Proposition 3.1 [31]. If a functional ϕ ∈ Esym is order continuous, then E ⊂ L1[0, ∞) and ϕ is proportional to the integral with respect to Lebesgue measure. 1048 A. L. Carey and F. A. Sukochev

We deﬁne the dilation operator Ds as in [42] by Dsf(t) = f(t/s). Note that Ds is ∗ ∗ a bounded operator on E and kDskE→E 6 max{1, s}; moreover, (Dsf) = Dsf for any function f ∈ E. The following result was established in [31] (Proposi- tion 2.3) under the assumption that ϕ is symmetric, but the proof holds also for rearrangement-invariant functionals.

∗ Proposition 3.2. If 0 6 ϕ ∈ E is rearrangement invariant, then ϕ(Dsf) = sϕ(f) for all f ∈ E and s > 0.

∗ 0 0 ∗ 0 A positive ϕ ∈ E is said to be singular if from 0 6 ϕ 6 ϕ, ϕ ∈ E , and ϕ being order-continuous it follows that ϕ0 = 0. Proposition 3.3 [31]. (i) Every symmetric functional on E can be uniquely decomposed into the sum of a normal functional and a singular symmetric functional. Moreover, the normal functional is zero unless E ⊆ L1[0, ∞). (ii) Any singular symmetric functional can be uniquely decomposed into a sum of singular symmetric functionals supported at zero and at inﬁnity. (iii) The set of symmetric functionals forms a lattice. The following result shows that every symmetric functional on E admits a ‘natural extension’ to a symmetric functional on E(N , τ) for any semiﬁnite von Neumann algebra (N , τ). By E(N , τ)+ we denote the set of all positive operators in E(N , τ). ∗ Theorem 3.4 [31]. Let ϕ0 ∈ Esym. If ϕ(x) := ϕ0(µ(x)) for all x ∈ E(N , τ)+, ∗ then ϕ extends to a symmetric functional 0 6 ϕ ∈ E(N , τ) .

Proof. It clearly suﬃces to show that ϕ is additive on E(N , τ)+. Let x, y ∈ E(N , τ)+. Since µ(x + y) ≺≺ µ(x) + µ(y) ([47], Theorem 4.4), and since ϕ0 is symmetric, it follows that

ϕ(x + y) = ϕ0(µ(x + y)) 6 ϕ0(µ(x) + µ(y)) = ϕ(x) + ϕ(y).

To prove the reverse inequality, we use the easily veriﬁed fact (see, for example, [33], Proposition 1.10) that

Z t Z t Z 2t µs(x) ds + µs(y) ds 6 µs(x + y) ds ∀ t > 0. (6) 0 0 0

Observing that (6) is equivalent to the submajorization

µ(x) + µ(y) ≺≺ 2D 1 µ(x + y), (7) 2 we get from Proposition 3.2 that

ϕ(x) + ϕ(y) = ϕ0(µ(x) + µ(y)) 2ϕ0(D 1 µ(x + y)) = ϕ0(µ(x + y)) = ϕ(x + y). 6 2

Thus, ϕ is additive on E(N , τ)+ and this suﬃces to complete the proof of the theorem. Dixmier traces and some applications 1049

Theorem 3.5. Suppose that (N , τ) is a semiﬁnite von Neumann algebra without minimal projections and E is a fully symmetric Banach function space on [0, ∞). ∗ If 0 6 ϕ ∈ E(N , τ) is a symmetric functional, then there exists a symmetric ∗ functional 0 6 ϕ0 ∈ E such that ϕ(x) = ϕ0(µ(x)) for all 0 6 x ∈ E(N , τ).

Proof. It is sufficient to show that there exists a symmetric functional 0 6 ϕ0 ∈ ∗ E[0, τ(1)) satisfying ϕ(x) = ϕ0(µ(x)) for all 0 6 x ∈ E(N , τ). Let M be the commutative von Neumann algebra L∞[0, τ(1)), with trace given by integration. The algebra Mfcan be identified with the space of all measurable functions on [0, τ(1)) which are bounded except on a set of finite measure. Since M does not con- tain any minimal projections, there exists a positive rearrangement-preserving algebra homomorphism J : Mf→ Nf ([51], Lemma 4.1, and [53], Theorem 3.5). Let ∗ 0 6 ϕ ∈ E(N , τ) be symmetric. For f ∈ E[0, τ(1)) we define ϕ0(f) := ϕ(Jf). ∗ It is clear that 0 6 ϕ0 ∈ E[0, τ(1)) is symmetric. Moreover, if 0 6 x ∈ E(N , τ), then µ(J(µ(x))) = µ(x), and hence ϕ(x) = ϕ(J(µ(x))) = ϕ0(µ(x)). This completes the proof of Theorem 3.5. It is even easier to see that the correspondence between the sets of symmetric functionals on E and on E(N , τ) given in Theorems 3.4 and 3.5 also exists for ∗ ∗ the sets (È)sym and (E(H ))sym of symmetric functionals. Furthermore, as the ∗ following result shows, there exists a simple connection between the sets (È)sym ∗ and Esym. Theorem 3.6 [31]. Suppose that E is a fully symmetric Banach function space on [0, ∞) and E(H ) is the corresponding ideal of compact operators on the infinite- ∗ dimensional Hilbert space H . If 0 6 ϕ ∈ E(N , τ) is a symmetric functional, then ∗ there exists a ϕ0 ∈ Esym such that ϕ(x) = ϕ0(µ(x)) for all x ∈ E(N , τ)+. Let us now consider the following question naturally arising from Theorem 3.4 and posed in [37]. Suppose that E (respectively, È) is a fully symmetric function (respectively, sequence) space and ϕ0 is a positive rearrangement-invariant functional on E (respectively, È). Is the functional

ϕ(x) := ϕ0(µ(x)) additive on E(N , τ) (respectively, E(H ))? Very recently this question was ans- wered in the aﬃrmative in [35], provided that N is a factor. The proof given in [35] is based on deep results in [32] (see also [54], [55]) concerning the structure of the commutator space [I,J] spanned by all the commutators [T,S], T ∈ I, S ∈ J (here, I and J are ideals of compact operators in L (H )). We remark that every rearrangement-invariant functional on the ideal I vanishes on [I, L (H )]. Many important results in [32], [54], [55] admit extension to the case of general semiﬁnite von Neumann algebras and their ideals [56], [57].

4. Preliminaries on dilation- and translation-invariant states

The construction of Dixmier traces τω depends crucially on the choice of the ‘invariant mean’ ω. In this section we recall and review the most important classes of such means. We denote by `∞ = `∞(N) the Banach space of all bounded sequences of complex numbers. By a state on a unital C∗-algebra we mean a positive linear 1050 A. L. Carey and F. A. Sukochev functional with value 1 on the identity of the algebra. We recall that a positive ∗ linear functional L ∈ `∞ is called a Banach limit if L is translation invariant and L(1) = 1 (here, 1 = (1, 1, 1,... )). A Banach limit L satisfies in particular L(ξ) = 0 for all ξ ∈ c0 (c0 is the set of all sequences in `∞ converging to zero). We denote the collection of all Banach limits on `∞ by BL(N). Note that kLk = 1 for all L ∈ BL(N). ∞ We recall that sequence ξ = {ξn}n=1 ∈ `∞ is said to be almost convergent to an α ∈ R (denoted by F -limn→∞ ξn = α) if and only if L(ξ) = α for all L ∈ BL(N). The notion of an almost convergent sequence is due to Lorentz [58], who showed ∞ that the sequence {ξn}n=1 is almost convergent to α if and only if the equality limp→∞(ξn + ξn+1 + ··· + ξn+p−1)/p = α holds uniformly for n = 1, 2,... . We denote by ac (respectively, ac0) the set of all almost convergent (respectively, almost convergent to 0) sequences in `∞. Clearly, ac and ac0 are closed subspaces of `∞. We define the shift operator T : `∞ → `∞, the Cesàro operator H : `∞ → `∞, and the dilation operators Dn : `∞ → `∞ for n ∈ N by the formulae

T (x1, x2, x3,... ) = (x2, x3, x4,... ), x + x x + x + x H(x , x , x ,... ) = x , 1 2 , 1 2 3 ,... , 1 2 3 1 2 3

Dn(x1, x2, x3,... ) = (x1, . . . , x1, x2, . . . , x2,... ) | {z } | {z } n n for all x = (x1, x2, x3,... ) ∈ `∞. Each of the above operators is positive and leaves invariant the identity element 1 of the algebra `∞ and consequently is bounded with norm equal to 1. Moreover, ∞ {Dn}n=1 is an Abelian semigroup. The main tool in our construction of various classes of invariant means is the well known Markov–Kakutani fixed point theorem. Theorem 4.1 (Markov–Kakutani). Let F be a locally convex Hausdorff space and let K be a non-empty compact convex subset of F . Let T be an Abelian semigroup of continuous linear operators on F such that S(K) ⊆ K for all S ∈ T . Then there exists an x ∈ K such that Sx = x for all S ∈ T . It is easy to see that the set of all fixed points in Theorem 4.1 forms a compact convex subset of K. We shall be applying the Markov–Kakutani theorem in the setting when F = ∗ ∗ ∗ ∗ ∗ (`∞) ,(L∞(R)) ,(L∞(R+)) equipped with the weak topology. For simplicity we present the proofs only for the first case. Lemma 4.2 ([41], [29]). The following statements hold: n (i) DnT = T Dn ∀ n > 1; (ii) HT x − T Hx ∈ c0 ∀ x ∈ `∞; (iii) HDnx − DnHx ∈ c0 ∀ x ∈ `∞. Proof. The proof of (i) is straightforward. For the proof of (ii) note that for all x ∈ `∞

1 x2 + ··· + xk+1 1 2 |(HT x)k − (T Hx)k| = − x1 kxk∞, k + 1 k k + 1 6 k + 1 which shows that HT x − T Hx ∈ c0. Dixmier traces and some applications 1051

We now prove (iii). Let n > 1 and x ∈ `∞. For 1 6 k ∈ N there exist l > 1 and 1 6 r 6 n such that k = (l − 1)n + r. Hence,

k l−1 1 X n X r (HD x) = (D x) = x + x n k k n i k j k l i=1 j=1 and l 1 X (D Hx) = (Hx) = x . n k l l j j=1 Noting that nl − k = n − r 6 n and rl − k 6 nl − k 6 n, we get that l−1 nl − k X rl − k |(HDnx)k − (DnHx)k| = xj + xl kl kl j=1 l−1 n1 X n 2n |x | + |x | kxk . 6 k l j k l 6 k ∞ j=1

(1,∞) This shows that HDnx − DnHx ∈ ` ⊆ c0, and Lemma 4.2 is proved.

Theorem 4.3. There exists a state ωe on `∞ such that for all n > 1

ωe ◦ T = ωe ◦ H = ωe ◦ Dn = ω.e ∗ ∗ Proof. Let K = {0 6 ϕ ∈ (`∞) : ϕ(1) = 1,T ϕ = ϕ}. Since K contains the usual Banach limits, it is not empty. It is clear that K is convex and weak∗-compact. We ∗ ∗ ∗ claim that Dn(K) ⊆ K. Indeed, by Lemma 4.2 (i) above we know that T Dn = ∗ ∗ n Dn(T ) , and hence for ϕ ∈ K ∗ ∗ ∗ ∗ n ∗ T (Dnϕ) = Dn(T ) ϕ = Dnϕ, ∗ which implies that Dnϕ ∈ K. Therefore, we may apply Theorem 4.1 to the set K ∗ ∞ and the Abelian semigroup {Dj }j=1. Consequently, the set ∗ ∗ ∗ K1 = {0 6 ϕ ∈ (`∞) : ϕ(1) = 1,T ϕ = ϕ, Dnϕ = ϕ, n > 1} ∗ is non-empty, and again it is clear that K1 is convex and weak -compact. ∗ We now show that H (K1) ⊆ K1. To this end, ﬁrst observe that ϕ(z) = 0 for ∗ all z ∈ c0 and all ϕ ∈ K1 (since T ϕ = ϕ). For a given ϕ ∈ K1 it follows from Lemma 4.2 (iii) that

∗ ∗ ∗ ∗ (DnH ϕ)(x) − (H Dnϕ)(x) = ϕ(HDnx − DnHx) = 0 for all x ∈ `∞, and so

∗ ∗ ∗ ∗ ∗ Dn(H ϕ) = H (Dnϕ) = H ϕ. ∗ ∗ ∗ Similarly, it follows from Lemma 4.2 (ii) that T (H ϕ) = H ϕ for all ϕ ∈ K1. Con- ∗ sequently, H (K1) ⊆ K1. Applying Theorem 4.1 to the set K1 and the semigroup ∗ n ∞ ∗ {(H ) }n=0, we can conclude that there exists an ωe ∈ K1 such that H (ωe) = ωe, which completes the proof. 1052 A. L. Carey and F. A. Sukochev

∗ The (multiplicative) group of positive real numbers is denoted by R+. ∗ We define an isomorphism L: L∞(R) → L∞(R+) by L(f) = f ◦ log. First, we ∗ define the Cesàromeans (transforms) on L∞(R) and L∞(R+), respectively, by 1 Z u H(f)(u) = f(v) dv for f ∈ L∞(R), u ∈ R, u 0 and, Z t 1 ds ∗ M(g)(t) = g(s) for g ∈ L∞(R+), t > 0. log t 1 s ∗ A brief calculation for g ∈ L∞(R+) gives us that 1 Z log r 1 Z r dv LHL−1(g)(r) = g(eu) du = g(v) = M(g)(r), log r 0 log r 1 v that is, L intertwines the two means. We shall now consider the analogues of the operators T , Dn and H acting ∗ on L∞(R) and L∞(R+).

Deﬁnition 4.4. Let Tb denote translation by b ∈ R, let Da denote dilation by ∗ a ∗ 1/a ∈ R+, and let P denote exponentiation to the power a ∈ R+. That is,

Tb(f)(x) = f(x + b) for f ∈ L∞(R), x D (f)(x) = f for f ∈ L ( ), a a ∞ R a a ∗ P (f)(x) = f(x ) for f ∈ L∞(R+).

Some of the basic relations between these L∞-spaces and their self-maps are provided for easy access by the following proposition, whose proof is similar to that of Lemma 4.2.

Proposition 4.5 [29]. The space L∞(R) together with the self-maps Da, Tb, and H ∗ (a > 0, b ∈ R) is connected with the space L∞(R+) together with its self-maps a P , Da, and M (a > 0) via the isomorphism ∗ L: L∞(R) → L∞(R+) and the following identities: −1 a (1) LD 1 L = P for a > 0, a −1 (2) LTbL = D(exp(b))−1 for b ∈ R, (3) LHL−1 = M, a a (4) DaH = HDa and P M = MP for a > 0, (5) limt→∞(HTb − TbH)f(t) = 0 for f ∈ L∞(R) and b ∈ R, ∗ (6) limt→∞(MDa − DaM)f(t) = 0 for f ∈ L∞(R+) and a > 0. Proposition 4.6 [29]. If a continuous functional ωe on L∞(R) is invariant under the Ces`aro operator H, the translation operator Ta, or the dilation operator Da, −1 ∗ then ωe ◦ L is a continuous functional on L∞(R+) invariant under M, the dila- a tion operator Da, or P , respectively. Conversely, composition with L converts a ∗ an M-, Da- or P -invariant continuous functional on L∞(R+) into an H-, Ta- or Da-invariant continuous functional on L∞(R). Dixmier traces and some applications 1053

∗ We denote by C0(R) (respectively, C0(R+)) the set of continuous functions on R ∗ (respectively, R+) vanishing at inﬁnity. The proof of the following theorem is analogous to that of Theorem 4.3.

Theorem 4.7 [29]. There exists a state ωe on L∞(R) satisfying the following conditions: (1) ωe(C0(R)) ≡ 0; (2) if f is real-valued in L∞(R), then

ess lim inf f(t) 6 ω(f) 6 ess lim sup f(t); t→∞ e t→∞

(3) if the essential support of f is compact, then ωe(f) = 0; (4) for all a > 0 and c ∈ R

ωe = ωe ◦ Tc = ωe ◦ Da = ωe ◦ H. The next result is a consequence of Theorem 4.7 and Proposition 4.6. ∗ Corollary 4.8. There exists a state ω on L∞(R+) satisfying the following conditions: ∗ (1) ω(C0(R+)) ≡ 0; ∗ (2) if f is real-valued in L∞(R+), then

ess lim inf f(t) 6 ω(f) 6 ess lim sup f(t); t→∞ t→∞ (3) if the essential support of f is compact, then ω(f) = 0; (4) for all a, c > 0 a ω = ω ◦ Dc = ω ◦ P = ω ◦ M. Theorem 4.7 and Corollary 4.8 enable one to develop an alternative approach to the theory of Dixmier [1] and Connes–Dixmier traces [7]. Whereas Dixmier’s original approach is based on the use of dilation-invariant functionals, we replace the latter with Banach limits (that is, translation-invariant functionals) and make use of the well-developed theory of almost convergent sequences. We introduce the following notation

BL(R) = {the set of all states ωe on L∞(R) satisfying conditions (1)–(3) of Theorem 4.7 and such that ωe ◦ Tc = ωe for all c ∈ R}, ∗ ∗ D(R+) = {the set of all states ω on L∞(R+) satisfying conditions (1)–(3) ∗ of Corollary 4.8 and such that ω ◦ Dc = ω for all c ∈ R+}, BL(R+) = {the set of all states ωe on L∞(R+) satisfying conditions (1)–(3) of Corollary 4.8 and such that ωe ◦ Tc = ωe for all c ∈ R+}. The following simple remark plays an important role in what is to follow. If ∗ −1 ω ∈ D(R+), then L := ω ◦ L belongs to BL(R). If L ∈ BL(R+), then ωe := L ◦ L ∗ ∗ belongs to D(R+). Finally, note that the isomorphism L: L∞(R) → L∞(R+) sends the space Cb(R) of all bounded continuous functions on R onto the space Cb(R+) of all bounded continuous functions on R+. Thus, one can reformulate all the results from Propositions 4.5 and 4.6, Theorem 4.7, and Corollary 4.8 for the spaces of bounded continuous functions on R and R+. 1054 A. L. Carey and F. A. Sukochev

5. Concrete constructions of singular symmetric functionals ∗ 5.1. Dixmier traces. If ω is a state on `∞ (respectively, on L∞(R), L∞(R+)), ∞ then we shall frequently denote its value on an element {xi}i=1 (respectively, ∗ f ∈ L∞(R), L∞(R+)) by ω-limi→∞ xi (respectively, ω-limt→∞ f(t)). Recall that Theorems 4.3 and 4.7 and Corollary 4.8 guarantee the existence of translation- ∗ and/or dilation-invariant states on `∞, L∞(R), L∞(R+). For simplicity we explain (1,∞) the construction of Dixmier traces for the ideal (L (H ), k · k(1,∞)) of compact operators deﬁned in § 2.1. (1,∞) Deﬁnition 5.1. A Dixmier trace of an operator T ∈ L+ (H ) is a number

N 1 X τω(T ) := ω-lim µn(T ), N→∞ log(1 + N) n=1 where ω is a D2-invariant state on `∞. Remark. We have deliberately chosen ω to satisfy only the dilation invariance assumption in the proof below, even though Dixmier originally imposed on ω the assumption of dilation and translation invariance. We shall discuss the dif- ferences below. (1,∞) Proposition 5.2. τω(S + T ) = τω(S) + τω(T ) for all S, T ∈ L+ (H ). Proof. For brevity we set

N X (1,∞) σN (X) := µn(X),X ∈ L (H ), n=1 and note that

σN (X) = sup{Tr(XP ): P = P (H ) is a projection and dim P (H ) = N}

(see, for example, [49], [47]). For a given ε > 0 let the projections P1 and P2 be such that dim P1(H ) = dim P2(H ) = N, Tr(SP1) > σN (S) − ε, and Tr(TP2) > σN (T ) − ε. Setting P := P1 ∨ P2, we have

Tr((S + T )P ) = Tr(SP ) + Tr(TP ) > Tr(SP1) + Tr(TP2) > σN (S) + σN (T ) − 2ε.

Since dim P (H ) 6 2N and ε is an arbitrary positive number, we have

σ2N (S + T ) > σN (S) + σN (T ),N > 1. Setting 1 1 1 α = Nσ (S), β = Nσ (T ), γ = Nσ (S + T ) N log(1 + t) N N log(1 + t) N N log(1 + t) N for brevity, we restate the above inequality as log(1 + 2N) γ α + β ,N 1. log(1 + N) 2N > N N > Dixmier traces and some applications 1055

Assume for a moment that we know that

ω-lim γ2N = ω-lim γN . (8) N→∞ N→∞ Noting that {γ } ∈ ` , so that (log(1 + 2N)/ log(1 + N))γ − γ → 0, we 2N N>1 ∞ 2N 2N infer from the above inequality that

τω(S + T ) = ω-lim γN ω-lim αN + ω-lim βN = τω(S) + τω(T ). N→∞ > N→∞ N→∞

Since the reverse inequality τω(S + T ) 6 τω(S) + τω(T ) follows immediately from the well-known inequality σN (S + T ) 6 σN (S) + σN (T ) (see [49], [47]), the proof is complete. It remains to establish the equality (8). The D2-invariance of ω immediately implies that ∞ ∞ ω({γ2N }N=1) = ω(D2{γ2N }N=1) = ω({γ2, γ2, γ4, γ4, γ6, γ6,... }), and therefore to prove (8) it is suﬃcient to verify that ∞ ∞ D2{γ2N }N=1 − {γN }N=1 = {γ2, γ2, γ4, γ4, γ6, γ6,... } − {γ1, γ2, γ3, γ4, γ5,... } ∈ c0 or, equivalently, that γ2N − γ2N−1 → 0 as N → ∞. For N > 2 we have 1 1 1 γ − γ = − σ (T + S) + · µ (T + S). 2N 2N−1 log 2N log(2N − 1) 2N−1 log 2N 2N It is obvious that the second term on the right-hand side tends to 0 as N → ∞. Noting that the condition T + S ∈ L (1,∞)(H ) guarantees the relations 1 1 1 σ (T +S) = Olog(2N −1) and − = o , 2N−1 log 2N log(2N − 1) log(2N − 1) we see that the ﬁrst term also tends to 0 as N → ∞. This completes the proof of Proposition 5.2.

∞ i P∞ Remark 5.3. The map {xj}j=1 7→ j=1 xjχ[j−1,j) deﬁnes an isometric embedding i: `∞ → L∞[0, ∞), where χ[j−1,j) is the characteristic function of [j − 1, j). We observe that if i({γN }N 1) = f, then i({γ2N }N 1) = f ◦ D 1 . Therefore, the proof > > 2 of (8) above would become trivial if ω were a D 1 -invariant state on L∞[0, ∞). 2 Deﬁnition 5.4. The Dixmier trace of a self-adjoint operator T ∈ L (1,∞)(H ) is τω(T ) := τω(T+) − τω(T−) and the Dixmier trace of an arbitrary operator T ∈ (1,∞) L (H ) is τω(T ) := τω(Re T ) + iτω(Im T ).

Proposition 5.5. The Dixmier trace τω is a (singular) symmetric functional on (1,∞) (1,∞) L (H ) such that τω(ST ) = τω(TS) for all T ∈ L (H ), S ∈ B(H ). (1,∞) Proof. The inequality |τω(x)| 6 kxk(1,∞), x ∈ L (H ), and then the fact that τω is a symmetric functional follow immediately from Definition 5.1 and Corol- lary 4.8. Since every operator S ∈ L (H ) is a linear combination of four unitary operators ([59], VI.6), it is sufficient to prove the equality τω(UT ) = τω(TU) for unitary U. Since every operator in L (H ) is a linear combination of positive operators, it is sufficient to prove the last equality for positive T . In this case the equality ∗ follows immediately from the fact that µn(UTU ) = µn(UT ) = µn(TU) = µn(T ) for all n > 1. Proposition 5.5 is proved. 1056 A. L. Carey and F. A. Sukochev

Definitions 5.1 and 5.4 extend to the Marcinkiewicz spaces L (1,∞)(N , τ), and further to the Marcinkiewicz spaces M(ψ)(N , τ), where ψ ∈ Ω satisfies the condi- ∗ tion (5). More precisely, let us fix an arbitrary state ω on L∞(R+) that satisfies the conditions (1)–(3) of Corollary 4.8 and is D 1 -invariant. Setting 2

τω(x) := ω-lim a(x, t), 0 x ∈ M(ψ)(N , τ), (9) t→∞ 6 and slightly modifying the argument (see the details in [31], p. 51) in the proofs of Propositions 5.2 and 5.5, we obtain an additive homogeneous functional on M(ψ)(N , τ)+ which extends to a symmetric functional on M(ψ)(N , τ) by linearity. ∗ Below we refer to any functional τω in (9) with ω ∈ D(R+) as a Dixmier trace. Finally, we remark that the duality between the dilation-invariant functionals ∗ on L∞(R+) and the translation-invariant functionals on L∞(R) makes possible an alternative definition of Dixmier traces. For simplicity we consider this definition only for the space L (1,∞)(N , τ), where (N , τ) is an arbitrary semifinite von Neu- mann algebra. Let L (respectively, L) belong to BL(R+) (respectively, BL(N)). We set

t 1 Z e FL(T ) := L-lim t µs(T ) ds, t→∞ log(1 + e ) 0 (1,∞) [eN ] T ∈ L (N , τ), 1 X FL(T ) := L-lim µn(T ), N→∞ log(1 + eN ) n=1 where [eN ] is the integral part of eN . Theorem 5.6 ([29], [40], [41], [60]). For any semifinite von Neumann algebra (N, τ) and arbitrary states L ∈ BL(R+) and L ∈ BL(N), the functionals FL and FL are symmetric functionals on L (1,∞)(N , τ). The following result shows that the class of Dixmier traces on L (1,∞)(N , τ) coincides with the sets {FL : L ∈ BL(N)} and {FL : L ∈ BL(R+)} of functionals. The proof of the first equality below follows from the remarks at the end of the preceding section. Theorem 5.7 ([60], Theorems 2.3, 6.2). For any semifinite von Neumann algebra (N , τ),

∗ {τω : ω ∈ D(R+)} = {FL : L ∈ BL(R+)} = {FL : L ∈ BL(N)}. A thorough study of the class of concave functions ψ ∈ Ω for which analogues of Theorems 5.6 and 5.7 hold for similarly deﬁned classes of symmetric functionals on the Marcinkiewicz spaces M(ψ)(N , τ) is contained in [40], [41], [60]. 5.2. Connes–Dixmier traces. We showed in the preceding subsection that with ∗ every state ω ∈ D(R+) (respectively, L ∈ BL(R+), BL(R), L ∈ BL(N)) there exists an associated Dixmier trace τω (respectively, symmetric functional FL, FL). It is ∗ possible to single out various subsets of the sets D(R+), BL(R+), BL(R), BL(N) of Dixmier traces and some applications 1057 states and to associate them with corresponding subsets of traces. For example, let us consider the sets of all H-invariant (respectively, M-invariant) states on L∞(R) ∗ or on `∞ (respectively, on L∞(R+)). It is easy to see that

{ω : ω is an H-invariant state on L∞(R) (respectively, on `∞)} $ {ω : ω ∈ BL(R) (respectively, BL(N))} (10) and ∗ ∗ {ω : ω is an M-invariant state on L∞(R+)} $ {ω : ω ∈ D(R+)}. (11) ∗ Indeed, suppose that 0 6 ω ∈ `∞ is such that ω(1) = 1 and ω(Hx) = ω(x) for all x ∈ `∞. To prove that ω(T x) = ω(x), x ∈ `∞, it is suﬃcient to show that ω(HT x) = ω(Hx), x ∈ `∞. However, a straightforward calculation yields x + ··· + x x + ··· + x x − x (HT x) − (Hx) = 2 N+1 − 1 N = N+1 1 → 0 N N N N N as N → ∞. Similarly, it can be shown that for all x ∈ L∞(R) and b ∈ R we have

lim ((HTbx)(t) − (Hx)(t)) = 0, t→∞ which establishes (10), and the inclusion (11) follows from (10) via Corollary 4.8. ∗ In [7] Connes proposed working with a set of states on L∞(R+) which is larger than the set on the left-hand side of (11). Namely, let us consider the following class of ∗ states on L∞(R+):

∗ CD(R+) := {ωe = γ ◦ M : γ is an arbitrary singular state on Cb[0, ∞)}. ∗ ∗ It is still easy to verify that CD(R+) $ D(R+) and then to deduce from Proposi- tion 4.5 the proper inclusion CBL(R+) $ BL(R+), where

CBL(R+) := {L = γ ◦ H : γ is an arbitrary singular state on Cb(R+)}. We refer to the subclass of Dixmier traces

∗ {τω : ω ∈ CD(R+)} as the class of Connes–Dixmier traces. The following theorem shows that an analogue of Theorem 5.7 also holds for the class of Connes–Dixmier traces. Theorem 5.8 ([60], Theorems 5.6, 6.2). For any semiﬁnite von Neumann algebra (N , τ), ∗ {τω : ω ∈ CD(R+)} = {FL : L ∈ CBL(R+)}. We complete this subsection with the remark that there is another natural sub- ∗ class of Dixmier traces which is associated with the subset of states on L∞(R+) appearing in Corollary 4.8, namely,

∗ ∗ a {ω ∈ L∞(R+) : ω is an M-invariant and P -invariant ∗ state on L∞(R+), a > 0} 1058 A. L. Carey and F. A. Sukochev or equivalently, with the set

∗ {L ∈ L∞(R) : L is an H-invariant and Da-invariant state on L∞(R), a > 0}. The class of Dixmier traces associated with the latter set will be called the class of maximally invariant Dixmier traces. Clearly, the latter class is contained in the class of Connes–Dixmier traces. Maximally invariant Dixmier traces are used in §§ 10 and 11. 5.3. Rearrangement-invariant functionals and singular traces. The class of Dixmier traces (and its subclasses) studied in the preceding subsections is a special subclass of the general class of (singular) symmetric functionals on the (fully symmetric) operator space L (1,∞)(N , τ) (or on the operator ideal L (1,∞)(H )). The latter class is, in turn, a subclass of the class of rearrangement-invariant functionals on L (1,∞)(N , τ). In the case of the operator ideal L (1,∞)(H ) the latter class may be viewed as the set of all (singular) traces on L (H ) with finite values on elements in L (1,∞)(H ). The literature devoted to general singular traces on L (H ) is extensive. We limit our list of papers from this area to [33], [34], [18]–[20], [61], [62]. In many cases the theory of singular traces runs in parallel to the theory of symmetric functionals. For instance, the results in [18] and [62] (respectively, [33]) concerning necessary and sufficient conditions on a positive compact operator T (respectively, a positive τ-measurable operator T ∈ L1(N , τ) + N ) for the existence of a singular trace taking a finite non-zero value on T are in fact very close to the result in Theorem 2.2. It should be noted that the results in [33] are stated for general τ-measurable operators, and not just for operators in L1(N , τ)+N , though not all the results there treating this general case are supplied with a reliable proof (see, for example, [33], Proposition 3.3). Finally, we point out the important connections between the theory of general singular traces and the study of the structure of commutator spaces for operators acting on a Hilbert space (see the comments at the end of § 3). The last investigations indicated above involve cyclic homology and algebraic K-theory for operator ideals and are beyond the scope of the present survey. For details, we recommend [32], [54], [56], [55], [57].

6. Class of measurable elements In this section we briefly review Connes’ notion of measurable operators [7]. Definition 6.1. An operator T ∈ L (1,∞)(H ) is said to be (Dixmier-) measurable ∗ if τω(T ) does not depend on the choice of ω ∈ D(R+). Definition 6.2 [7]. An operator T ∈ L (1,∞)(H ) is said to be Connes–Dixmier- ∗ measurable if τω(T ) does not depend on the choice of ω ∈ CD(R+). Remark 6.3. (i) It is obvious that the sets of measurable operators defined above are linear spaces which are, in fact, closed subspaces of L (1,∞)(H ). However, these subspaces are not order ideals, in other words, the fact that a self-adjoint operator A is measurable does not necessarily imply that A+ and A− are measurable. Consider the following example. Take a positive non-measurable diagonal Dixmier traces and some applications 1059

(1,∞) operator A = diag{a1, a2, a3,... } in L (H ) and define a diagonal operator B by B = diag{a1, −a1, a2, −a2,... }. Obviously, B is measurable, and moreover τω(B) = 0 for all ω. However, the positive and negative parts of B are not measurable. (ii) The definitions of Dixmier- and Connes–Dixmier-measurable operators extend naturally [60] to the Marcinkiewicz spaces L (1,∞)(N , τ) with (N , τ) an arbitrary semifinite von Neumann algebra, and further, to the operator Marcinkiewicz spaces M(ψ)(N , τ) for all ψ satisfying condition (3). Clearly, Dixmier-measurable operators are Connes–Dixmier-measurable. In this section our object is to describe the classes of positive Connes–Dixmier-measurable operators and positive Dixmier-measurable operators and to show that these two classes actually coincide. To this end, we will need two auxiliary results. Theorem 6.4 ([63], § 6.8). Let b(t) be a positive piecewise differentiable function such that tb0(t) > −H for some H > 0 and all t > C, where C is a constant. Then

1 Z t lim b(s) ds = A for some A > 0 if and only if lim b(t) = A. t→∞ t 0 t→∞ Imitating the Lorentz deﬁnition of almost convergent sequences, we say that a positive function f ∈ Cb[0, ∞) is almost convergent if the states in BL(R+) all take the same value on this function.

Theorem 6.5 ([60], Theorem 3.3). If a function f ∈ Cb[0, ∞) is almost convergent to a number A, then the limit

1 Z t lim f(s) ds t→∞ t 0 exists and is equal to A.

Proof. Suppose not. Then there exists a constant c 6= A such that (Hf)(tn) → c ∗ for some sequence tn ↑ ∞. Let B be the unit ball of Cb([0, ∞)) and consider ∗ the sequence of functionals σtn (x) = x(tn), n > 1, in B. Since B is weak -compact, this sequence has a limit point V ∈ B. It is easy to see that V > 0, V (1) = 1, V (p)=limn→∞ p(tn)=0 for all p∈C0[0, ∞), and V (H(f)) = limn→∞ H(f)(tn) = c. We deﬁne a functional L on L∞(R+) by L(x) := V (H(x)). It is easy to verify that L is a state in BL(R+) and V (f) = c 6= A. Thus, the supposition that the result does not hold is false. The following theorem is the main result of this section. Theorem 6.6 [60]. A positive operator T in L (1,∞)(N , τ) is Dixmier-measurable if and only if the limit 1 Z t lim µs(T ) ds t→∞ log(1 + t) 0 exists. (1,∞) Proof. The ‘if’ part is trivial. Let T ∈ L+ (N , τ) be a ﬁxed operator such 1 R t that for g(t) := log(1+t) 0 µs(T ) ds we have τω(T ) = ω-limt→∞ g(t) = A > 0 1060 A. L. Carey and F. A. Sukochev

∗ for all ω ∈ CD(R+). It follows from the remarks made in § 5.2 that TrL(T ) := λ L-limλ→∞ g(e ) = A for all L ∈ BL(R+), and therefore, by Theorem 6.5,

λ 1 Z u 1 Z e lim λ µs(T ) ds dλ = A. u→∞ u 0 log(1 + e ) 0 Setting λ 1 Z e b(λ) := λ µs(T ) ds, λ > 0, log(1 + e ) 0 we have

λ λ d 1 Z e λeλ Z e λb0(λ) λ µ (T ) ds = − µ (T ) ds > λ s λ 2 λ s dλ log(1 + e ) 0 (1 + e ) log (1 + e ) 0 λ λ 1 Z e > − λ · λ µs(T ) ds > −kT k(1,∞). log(1 + e ) log(1 + e ) 0

By Theorem 6.4 we now deduce that limλ→∞ b(λ) = A, and hence limt→∞ g(t) = A. This completes the proof of Theorem 6.6. We shall now show that an argument similar to that in the proof above yields a stronger result. Theorem 6.7 [60]. A positive operator T in L (1,∞)(N , τ) is Connes–Dixmier- measurable if and only if the limit

1 Z t lim µs(T ) ds t→∞ log(1 + t) 0 exists. Proof. We need only show the ‘only if’ part. We use the notations g( · ) and b( · ) (1,∞) in the proof of Theorem 6.6. Suppose that T ∈ L+ (N , τ) satisﬁes the equality −1 ∗ A = γ ◦ M(g) = γ ◦ LHL (g) for any state γ ∈ Cb [0, ∞) vanishing on C0(0, ∞). Note that if γ is a dilation-invariant state, then γ ◦ L is a translation-invariant ∗ state. This remark (and the fact that L: L∞(R) → L∞(R+) is an isomorphism) shows that HL−1(g) = Hb is almost convergent. Applying Theorem 6.5 to the function Hb, we see that the limit limt→∞ HH(b)(t) exists. Assume for a moment that we have already veriﬁed the assumption of Theorem 6.4 for the function Hb. Then we infer from that theorem that the limit limt→∞ H(b)(t) also exists, and, using the same theorem again (as in the proof of Theorem 6.6), we conclude that limt→∞ b(t) exists, and hence so does limλ→∞ g(λ). It remains to verify the assumption of Theorem 6.4 for Hb. For all λ > 1 we have λb(λ) − R λ b(s) ds λ(Hb)0(λ) = 0 −kbk −kT k . λ > ∞ > (1,∞) Theorem 6.7 is proved. Corollary 6.8. The set of all positive Dixmier-measurable operators and the set of all positive Connes–Dixmier-measurable operators coincide. Dixmier traces and some applications 1061

The following questions are open: (i) Do the spaces of Dixmier-measurable and Connes–Dixmier-measurable operators coincide? (ii) What is the description of the set of all (positive) operators which are measurable with respect to the set of all maximally invariant Dixmier traces? (iii) What is the description of the set of all (positive) operators measurable with respect to the set of all symmetric functionals? We might be tempted to pose question (iii) in greater generality and ask whether the concept of operators measurable with respect to the set of rearrangement-invariant functionals can be sensibly formulated. There is an example in [35] (see the end of § 2) which answers this in the negative. We refer the reader to [60] for results extending Theorems 6.6 and 6.7 to the Marcinkiewicz spaces M(ψ)(N , τ) with ψ ∈ Ω satisfying condition (4).

7. Norming properties of Dixmier and Connes–Dixmier functionals The reader may have the impression that Dixmier and Connes–Dixmier traces form a very ‘thin’ subset of the unit sphere of the dual space. This impression is wrong, as established by Theorems 7.3 and 7.4 below. We shall need the following theorem of Sucheston.

Theorem 7.1 [64]. For x ∈ `∞

n 1 X sup L(x) = lim sup xm+j . n→∞ m n L∈BL(N) j=1

The next proposition follows easily from its ‘commutative’ counterpart, which in turn can be obtained from [42]. Proposition 7.2. Let T ∈ L (1,∞)(N , τ). The distance from T to the subspace (1,∞) L0 (N , τ) in the norm k · k(1,∞) is equal to

1 Z t ρ(T ) := lim sup µs(T ) ds. t→∞ log(1 + t) 0

(1,∞) We recall that L0 (H ) is the closed linear span of the set of finite-dimensional operators in L (1,∞)(H ) in the special case when L (1,∞)(N , τ) = L (1,∞)(H ). Theorem 7.3 [60]. Let T ∈ L (1,∞)(N , τ). The distance from T to the subspace (1,∞) ∗ L0 (N , τ) in the norm k · k(1,∞) is equal to sup{τω(|T |): ω ∈ D(R+)}. Proof. It is sufficient to consider the case T > 0. We note first that

∗ sup{τω(T ): ω ∈ D(R+)} = sup{TrL(T ): L ∈ BL(R+)} = sup{TrL(T ): L ∈ BL(N)}. It is clear that

1 Z t q(T ) := sup Tr (T ) lim sup µ (T ) ds (= ρ(T )). L 6 log(1 + t) s L∈BL(N) t→∞ 0 1062 A. L. Carey and F. A. Sukochev

We have to prove the reverse inequality q(T ) > ρ(T ). By Theorem 7.1 of Sucheston, it is enough to prove that

n m+j 1 X 1 Z e ∀ ε > 0 ∃ N ∈ : ∀ n N ∃ m ∈ : µ (T ) ds ρ(T ) − ε. N > N n m + j s > j=1 0

For this, it is enough to put N = 1 and take m such that

m m ρ(T ) − ε 1 Z e > and m µs(T ) ds > ρ(T ) − ε/2. (12) m + n ρ(T ) − ε/2 log(1 + e ) 0 Then

n m+j n m 1 X 1 Z e 1 X 1 Z e µ (T ) ds µ (T ) ds n m + j s > n m + n s j=1 0 j=1 0 m m 1 Z e = · µs(T ) ds > ρ(T ) − ε. m + n m 0

To verify that m can be chosen to satisfy (12), we ﬁrst ﬁnd a sequence 1 6 t1 < t2 < · · · ↑ ∞, such that

1 Z tk lim µs(T ) ds > ρ(T ) − ε/4, k > 1 (13) k→∞ log(1 + tk) 0

m −1 (this is possible by Proposition 7.2). For each k we deﬁne mk ∈ N so that e k 6 m tk 6 e k . Then

m 1 Z tk 1 Z e k µs(T ) ds 6 m −1 µs(T ) ds log(1 + tk) 0 log(1 + e k ) 0 mk log(1 + emk ) 1 Z e = m −1 · m µs(T ) ds. log(1 + e k ) log(1 + e k ) 0

Since log(1 + emk )/ log(1 + emk−1) → 1, we see that (12) follows from (13). This completes the proof of Theorem 7.3. A natural question is whether the norming property remains true for the class of Connes–Dixmier traces. The answer is given below. Theorem 7.4 [60]. Let T ∈ L (1,∞)(N , τ). The distance from T to the subspace (1,∞) L0 (N , τ) in the norm k · k(1,∞) is equivalent to supω τω(T ), where the supre- mum is taken over all singular states ω = γ ◦M with γ a singular state on Cb[0, ∞). As in the preceding section, the results given in Theorems 7.3 and 7.4 can be generalized to the Marcinkiewicz spaces M(ψ)(N , τ) with ψ ∈ Ω satisfying condition (4). We finish this section with the comment that it is not yet clear whether the difference in the results in Theorems 7.3 and 7.4 means that the set of all Dixmier traces is different from the set of all Connes–Dixmier traces. Dixmier traces and some applications 1063

8. Fredholm modules and spectral triples 8.1. Notation and definitions. Let N be a semifinite von Neumann algebra on a separable Hilbert space H , and let Lp(N , τ) be the non-commutative Lp-space associated with (N , τ), where τ is a faithful, normal semifinite trace on N . Let A be a unital Banach ∗-algebra represented in N via a continuous ∗-homomorphism π, which we can assume without loss of generality to be faithful. Where no confusion can arise, we suppress π in the notation. The fundamental objects of our analysis are explained in the following definition.

Let KN be the τ-compact operators in N (that is, the norm closed ideal gen- erated by the projections E ∈ N with τ(E) < ∞).

Definition 8.1. (i) A semifinite odd spectral triple (A , H , D) is given by a Hilbert space H , a ∗-algebra A ⊂ N with N a semifinite von Neumann algebra acting on H , and a densely defined unbounded self-adjoint operator D affiliated with N such that: 1) [D, a] is densely defined and extends to a bounded operator for all a ∈ A ; −1 2) (λ − D) ∈ KN for all λ∈ / R. (ii) We say that (A , H , D) is even if in addition there is a Z2-grading such that A is even and D is odd, that is, there is an operator Γ such that Γ = Γ∗, Γ2 = 1, Γa = aΓ for all a ∈ A , and DΓ + ΓD = 0.

(iii) If I is a symmetrically normed ideal in KN , then we say that the spectral triple (A , N , D) is I -summable if (1 + D 2)−1/2 ∈ I .

Remark 8.2. In [22] the terminology for the concept just introduced is von Neu- mann spectral triple. The two most important special cases of this deﬁnition are when: a) I is the ideal Lp, in which case we say that (A , N , D) is p-summable; b) the case where the Dixmier trace ﬁrst arose, namely, I = L p,∞, in which case we say that (A , N , D) is (p, ∞)-summable. There is a third case which is not relevant for the discussion here: the notion of θ-summability [65], [66], [7].

Remark. For simplicity, we deal in this paper only with unital algebras A ⊂ N in which the identity of A is that of N . We have adopted a notational convention corresponding to the context in which we are working. A script D will always denote an unbounded self-adjoint operator forming part of a semiﬁnite spectral triple (A , H , D). A roman D will denote a self-adjoint operator acting in a Hilbert space, usually with some side conditions.

In the original paper on non-commutative geometry by Connes [67] the notion of spectral triple was introduced as an ‘unbounded Fredholm module’ (see also [7]). The study of semifinite spectral triples was initiated in [23] in the context of spectral flow [68], [69] and was carried further in [22], with a focus on applications to foliations. In [67] the notion of Fredholm module was introduced, based on the concept of a Kasparov module. This can be generalized to the semifinite case as follows. 1064 A. L. Carey and F. A. Sukochev

Definition 8.3 [23], [39]. A bounded p-summable pre-Breuer–Fredholm module for A is a pair (N ,F0), where F0 is a bounded self-adjoint operator in N such that: 2 1/2 (1) |1 − F0 | belongs to Lp(N , τ); (2) Ap := {a ∈ A :[F0, a] ∈ Lp(N , τ)} is a dense ∗-subalgebra of A . 2 When F0 = 1 we drop the prefix pre. In the special case when N = L (H ) and τ is the standard trace Tr, we omit the name Breuer from the definition. In this case the non-commutative Lp-space coincides with the Schatten–von Neumann ideal Cp of compact operators, and the definition coincides with Definition 3 in [7], p. 290. 8.2. Bounded versus unbounded modules. The relationship between the bounded and unbounded pictures is worth some comment. There are two approa- ches to this question, that in [70] (and explained in more detail in [8]) and a more general approach via perturbation theory in [28]. In the case of even spectral triples this matter was settled by Connes (see [67], I.6). Let D and D0 be unbounded self-adjoint operators acting in H and differing by a bounded operator in N . We study directly the map ϕ defined by ϕ(D) = D(1 + D2)−1/2 and the difference ϕ(D) − ϕ(D0). In [28] a number of results were established, the one most relevant to this survey being the following.

Theorem 8.4. With the assumptions above about D and D0 and with D−D0 ∈ N and 1 < p < ∞,

1/2 2 −1/2 kϕ(D)−ϕ(D0)kLp(N ,τ) 6 Zp max{kD −D0k , kD −D0k}·k(1+D ) kLp(N ,τ) for some positive constant Zp which depends only on p. From this theorem (see [70]) one obtains the following corollary.

Corollary 8.5. Suppose that 1 < p < ∞ and (A , N , D0) is an odd semifinite p-summable spectral triple for a Banach ∗-algebra. Then (N , sgn(D0)) is an odd bounded p-summable Breuer–Fredholm module for A . 8.3. More on semifinite spectral triples. Definition 8.6. A semifinite spectral triple (A , H , D) belongs to the class QCk for k > 1 (Q for quantum) if for all a ∈ A the operators a and [D, a] are in the domain of δk, where δ(T ) = [|D|,T ] is the partial derivation on N defined by |D|. Remark. (i) The notation is meant to be analogous to the classical case, but we introduce the Q so that there is no confusion between quantum differentiability of a ∈ A and classical differentiability of functions. (ii) By a partial derivation we mean that δ is defined on some subalgebra of N which need not be (weakly) dense in N . More precisely, dom δ = {T ∈ N : δ(T ) is bounded}. Observation. If T ∈ N , then one can show that [|D|,T ] is bounded if and only if [(1 + D 2)1/2,T ] is bounded, by using the functional calculus to show 2 1/2 that |D| − (1 + D ) extends to a bounded operator in N . In fact, writing |D|1 = 2 1/2 n n (1 + D ) and δ1(T ) = [|D|1,T ], we have dom δ = dom δ1 for all n. Dixmier traces and some applications 1065

Proof. Let f(D) = (1 + D 2)1/2 − |D|. Then, as noted above, f(D) extends to a bounded operator in N . Since the operator

δ1(T ) − δ(T ) = [f(D),T ] is always bounded, dom δ = dom δ1. Now δδ1 = δ1δ, therefore, 2 2 δ1(T ) − δ (T ) = δ1(δ1(T )) − δ1(δ(T )) + δ1(δ(T )) − δ(δ(T )) = [f(D), δ1(T )] + [f(D), δ(T )]. 2 2 Both terms on the right-hand side are bounded, so dom δ = dom δ1. The proof proceeds by induction. Thus, the condition deﬁning the class QCk can be replaced by

\ n a, [D, a] ∈ dom δ1 ∀ a ∈ A . 06k6n This is important, since we do not assume at any point that |D| is invertible. If (A , H , D) is a QCk spectral triple, then we can endow the algebra A with the topology determined by the seminorms a → kδn(a)k + kδn([D, a])k, n = 0, 1, . . . , k. By Lemma 16 in [71] we can suppose without loss of generality that A is complete in the resulting topology, by completing it if necessary. This completion is stable under the holomorphic functional calculus, so we have a meaningful spectral theory, ∼ ∗ and K∗(A ) = K∗(A ) via inclusion, where A is the C -completion of A .A QC∞ spectral triple is one that is QCk for all k = 1, 2,... . Observation. If T ∈ N and [D,T ] is bounded, then [D,T ] ∈ N . Proof. We observe that D is affiliated with N , and so commutes with all projections in the commutant of N , and the commutant of N preserves the domain of D. Thus if [D,T ] is bounded, then it too commutes with all the projections in the commutant of N , and these projections preserve the domain of D, so that [D,T ] ∈ N . Similar remarks apply to [|D|,T ], [(1 + D 2)1/2,T ], and combinations such as [D 2,T ](1 + D 2)−1/2. We will often write L 1 for the ideal of trace-class operators 1 in order to simplify the notation, and we denote the norm on L by k · k1. We remark that L 1 need not be complete in this norm in the case when N 6= L (H ), but it is complete in the norm k · k1 + k · k∞, where k · k∞ is the uniform norm. 8.4. Summability and dimension. The finite summability condition on a spectral triple gives a half-plane in which the function z 7→ τ((1 + D 2)−z) (14) is well defined and holomorphic. Definition 8.7. If (A , H , D) is a QC∞ spectral triple, then we call 2 −q/2 p = inf{q ∈ R : τ((1 + D ) ) < ∞} the spectral dimension of (A , H , D). 1066 A. L. Carey and F. A. Sukochev

9. Spectral flow One of the main motivations for extending the study of spectral triples to the setting of semifinite von Neumann algebras is the investigation of type II spectral flow (this concept is due to Phillips [72], [73]). Let π : N → N /KN be the canonical map. A Breuer–Fredholm operator is one that maps to an invertible operator under π. A full discussion of Breuer–Fredholm theory in a semifinite von Neumann algebra is contained in [26], extending the discussion in the Appendix of [74] and in [75], [76]. As usual, D is an unbounded densely defined self-adjoint Breuer–Fredholm operator acting in H (in the sense that D(1+D2)−1/2 is bounded 2 −1/2 and is a Breuer–Fredholm operator in N ) with (1 + D ) ∈ KN . For a unitary u ∈ N such that [D, u] is a bounded operator, the path

u ∗ Dt := (1 − t)D + tuDu of unbounded self-adjoint Breuer–Fredholm operators is continuous in the sense that 1 u u u 2 − 2 Ft := Dt (1 + (Dt ) ) is a norm-continuous path of self-adjoint Breuer–Fredholm operators in N [23]. We recall that the Breuer-Fredholm index of a Breuer–Fredholm operator F is deﬁned by ind(F ) = τ(Qker F ) − τ(Qcoker F ), where Qker F and Qcoker F are the projections onto the kernel and cokernel of F .

Definition 9.1. If {Ft} is a continuous path of self-adjoint Breuer–Fredholm operators in N , then the definition of the spectral flow sf({Ft}) of the path is based on the following sequence of observations in [77], [78], and [73] (see also [79] and [80]). 1. For Ft = 2Pt − 1 with Pt the non-negative spectral projection, the function t 7→ sgn(Ft) is typically discontinuous, as is the projection-valued map t 7→ Pt = 1 2 (sgn(Ft) + 1), but t 7→ π(Pt) is continuous. 2. For projections P and Q in N , PQ: QH → P H is a Breuer–Fredholm operator if and only if kπ(P ) − π(Q)k < 1, in which case ind(PQ) ∈ R is well defined. 3. If we partition the parameter interval of {Ft} so that the π(Pt) do not vary much in norm on each subinterval of the partition, then

n X sf({Ft}) := ind(Pti−1 Pti ) i=1 is a well-defined and (path-) homotopy-invariant number which agrees with the usual notion of spectral flow in the type I∞ case. u 4. For D and u as above, we define the spectral flow of the path Dt := (1−t)D+ ∗ u u 2 − 1 tuDu to be the spectral flow of the path Ft, where Ft = Dt (1 + (Dt ) ) 2 . We denote this by ∗ sf(D, uDu ) = sf({Ft}) and observe that this is an integer in the N = L (H ) case and a real number in the general semifinite case. Dixmier traces and some applications 1067

Special cases of spectral flow in a semifinite von Neumann algebra were discussed in [81], [77], [78], [82], [72], [73]. Let P denote the projection onto the non-negative spectral subspace of D. The u spectral flow along {Dt } is equal to sf({Ft}), and this is the Breuer–Fredholm index ∗ u ∗ ∗ of P uP u by [23]. (Note that sgn F1 = 2uP u − 1 and that P − uP u is compact for this special path, and thus P uP u∗ is certainly a Breuer–Fredholm operator from uP u∗H to P H .) Then ([74], Appendix B) ind(P uP u∗) = ind(P uP ). The operator P uP is known as a generalized Toeplitz operator. Formulae for its index in terms of the Dixmier trace are discussed in §§ 13 and 16. These Dixmier trace formulae follow from the analytic formulae for spectral flow (discovered in [83], [23], [24]), which we will now explain. 9.1. Spectral flow formulae. We now introduce the spectral flow formula of Carey and Phillips [23], [24]. This formula starts with a semifinite spectral triple (A , H , D) and computes the spectral flow from D to uDu∗, where u ∈ A is unitary with [D, u] bounded, in the case where (A , H , D) has dimension p > 1. Thus, for any n > p we have

1 Z 1 sf(D, uDu∗) = τ(u[D, u∗](1 + (D + tu[D, u∗])2)−n/2) dt (15) Cn/2 0 with Z ∞ 2 −n/2 Cn/2 = (1 + x ) dx, −∞ by the extension of Theorem 9.3 in [24] to the case of general semifinite von Neu- mann algebras (see [27]). This real number sf(D, uDu∗) recovers the pairing of the K-homology class [D] of A with the K1(A ) class [u] (see below). There is a geometric way to view this formula, due originally to Getzler [83]. It is shown in [24] 2 −n/2 ∗ that the functional X 7→ τ(X(1 + (D + X) ) ) on Nsa = {T ∈ N : T = T } determines an exact one-form on an affine space D +Nsa. Thus, (15) represents the ∗ integral of this 1-form along the path {Dt = (1 − t)D + tuDu } if one understands ˙ ∗ Dt = u[D, u ] as the tangent vector to this path. Moreover, this formula is scale invariant. By this we mean that if we replace D by εD for ε > 0 in the right-hand side of (15), then the left-hand side is unchanged, as is evident from the definition of spectral flow. This is because spectral flow only involves the phase of D which is the same as the phase of εD. 9.2. Relation to cyclic cohomology. To place some of the discussion in its correct context and to prepare for later applications, we need to discuss some aspects of cyclic cohomology. We will use the normalized (b, B)-bicomplex (see [7], [84]). ⊗m We introduce the following linear spaces. Let Cm = A ⊗ A , where A is the quotient A /CI (with I the identity element of A ) and (assuming without loss of generality that A is complete in the δ-topology) we employ the projective tensor m product. Let C = Hom(Cm, C) be the linear space of continuous multilinear functionals on Cm. We can define the (b, B)-bicomplex using these spaces (as opposed ⊗m+1 to Cm = A , and so on), and the resulting cohomology will be the same. This ⊗m follows because the bicomplex defined using A ⊗ A is quasi-isomorphic to that defined using A ⊗ A ⊗m. 1068 A. L. Carey and F. A. Sukochev

A normalized (b, B)-cochain ϕ is a ﬁnite collection of continuous multilinear functionals on A :

m ϕ = {ϕm}m=1,...,M with ϕm ∈ C .

m It is a (normalized) (b, B)-cocycle if bϕm + Bϕm+2 = 0 for all m, where b: C → Cm+1 and B : Cm → Cm−1 are the coboundary operators given by

m−1 X (m−1)j (Bϕm)(a0, a1, . . . , am−1) = (−1) ϕm(1, aj, aj+1, . . . , am−1, a0, . . . , aj−1), j=0 m−2 X j (bϕm−2)(a0, a1, . . . , am−1) = (−1) ϕm−2(a0, a1, . . . , ajaj+1, . . . , am−1) (16) j=0 m−1 + (−1) ϕm−2(am−1a0, a1, . . . , am−2).

We write (b + B)ϕ = 0 for brevity. Thought of as functionals on A ⊗m+1, a normalized cocycle will satisfy the condition ϕ(a0, a1, . . . , an) = 0 whenever aj = 1 for at least one j > 1. An odd (even) cochain is one such that {ϕm} = 0 for m even (odd). T T Similarly, a (b ,B )-chain c is a (possibly infinite) collection c = {cm}m=1,2,... T T T T with cm ∈ Cm.A(b, B)-chain {cm} is a (b ,B )-cycle if b cm+2 + B cm = 0 for all m. More briefly, we write (bT + BT )c = 0. Here bT and BT are the boundary operators of the cyclic homology and are the transposes of the coboundary operators b and B in the following sense. M T T The pairing between a (b, B)-cochain ϕ = {ϕm}m=1 and a (b ,B )-chain c = {cm} is given by M X hϕ, ci = ϕm(cm). m=1 This pairing satisfies the condition

h(b + B)ϕ, ci = hϕ, (bT + BT )ci.

We have the relations

b2 = B2 = 0 = bB + Bb = (b + B)2, and thus we can define the cyclic cohomology of A as the cohomology of the total (b, B)-complex. In this survey our main application of the Dixmier trace to cyclic cohomology is the discussion in § 17 of Connes’ formula for the Hochschild class of the Chern character. Here we use the unrenormalized complex Cem = A ⊗m+1. The definition of the Hochschild coboundary b on Cem involves the same formula (16). The Hochschild cohomology, denoted by HH∗(A , A ∗), is then the cohomology of the complex (Ce∗(A ), b). ⊗m+1 T The Hochshild boundary on the complex Cem = A is the operator b . We say that c is a Hochschild cycle if bT c = 0. When the Hochschild homology is well defined, we denote it by HH∗(A ). Dixmier traces and some applications 1069

One can interpret spectral ﬂow (in the type I case) as the pairing between an odd K-theory class represented by a unitary u, and an odd K-homology class represented by (A , H , D) ([7], Chapters III, IV). This point of view also makes sense in the general semiﬁnite setting, though one must suitably interpret K-homology ([25], [24]). A central feature of [7] is the translation of the K-theory pairing to the cyclic theory in order to obtain index theorems. One associates to a suitable representative of a K-theory class (respectively, a K-homology class) a class in periodic cyclic homology (respectively, a class in periodic cyclic cohomology) called a Chern character in both cases. (We will not digress here to discuss the periodic theory and the periodicity operator, referring instead to [7] and [8].) The principal result is then

∗ 1 ∗ sf(D, uDu ) = h[u], [(A , H , D)]i = −√ h[Ch∗(u)], [Ch (A , H , D)]i, (17) 2πi where [u] ∈ K1(A ) is the K-theory class with representative u and [(A , H , D)] is the K-homology class of the spectral triple (A , H , D). On the right-hand side, Ch∗(u) is the Chern character of u. We recall that the Chern character of a unitary u is the following (inﬁnite) collection of odd chains Ch2j+1(u) satisfying b Ch2j+3(u) + B Ch2j+1(u) = 0: j ∗ ∗ Ch2j+1(u) = (−1) j! u ⊗ u ⊗ · · · ⊗ u ⊗ u . | {z } 2j + 2 entries

We have used the notation [Ch∗(u)] for the periodic cyclic homology class. Similarly, [Ch∗(A , H , D)] is the periodic cyclic cohomology class of the Chern character of (A , H , D).

10. The Dixmier trace and residues of the zeta function Many applications of the Dixmier trace are based on the possibility of calculating it by taking a residue of an associated zeta function. The main result in this direction in the type I case is Proposition IV.2.4 in [7]. Recent advances [29] have extended this to the general case of semifinite von Neumann algebras. 10.1. Preliminaries. First, it is useful to have an estimate of the singular values of the operators in L (1,∞). Lemma 10.1. For T ∈ L (1,∞) positive there is a constant K > 0 such that for each p > 1 Z t Z t p p 1 µs(T ) ds 6 K p ds. 0 0 (s + 1) Proof. By [47] (Lemma 2.5 (iv)), for all 0 6 T ∈ N and all continuous increasing functions f on [0, ∞) with f(0) > 0 we have µs(f(T )) = f(µs(T )) for all s > 0. Combining this fact with the well-known Hardy–Littlewood–Pólya result (see, for example, [48], Lemma 4.1), we see that if T1 ≺≺ T2 and 0 6 T1, T2 ∈ N , then p p (1,∞) T1 ≺≺ T2 for all p ∈ (1, ∞). By the definition of L , the singular values of T R t satisfy the condition 0 µs(T ) ds = O(log t), and hence for some K > 0 Z t Z t 1 µs(T ) ds 6 K ds ∀ t > 0. 0 0 (s + 1) 1070 A. L. Carey and F. A. Sukochev

In other words, µs(T ) ≺≺ K/(1 + s), and the assertion of the lemma follows immediately. In the next theorem we use the dilation-invariant states in Theorem 4.7. ∗ ∗ Theorem 10.2 (weak Karamata theorem). Assume that ωe ∈ L∞(R) is a dilation- invariant state and let β be a real-valued, increasing, right-continuous function on R+ which is zero at zero and such that the integral Z ∞ − t h(r) = e r dβ(t) 0 converges for all r > 0 and the limit C = ω^-limr→∞ h(r)/r exists. Then 1 β(t) ω^-lim h(r) = ω^-lim . r→∞ r t→∞ t Remark 10.3. The classical Karamata theorem states, in the notation of the theorem, that if the ordinary limit limr→∞ h(r)/r = C exists, then C = limt→∞ β(t)/t. The proof of this classical result is obtained by replacing ω^-lim everywhere in the proof of Theorem 10.2 by the ordinary limit. Proof. Let ( −1 −1 x for e 6 x 6 1, g(x) = −1 0 for 0 6 x < e , so that g is right-continuous at e−1. Then for r > 0 the function t 7→ e−t/rg(e−t/r) is left-continuous at t = r. Thus, the Riemann–Stieltjes integral Z ∞ e−t/rg(e−t/r) dβ(t) 0 exists for any r > 0. We assert that for any polynomial p 1 Z ∞ Z ∞ ω^-lim e−t/rp(e−t/r) dβ(t) = C e−tp(e−t) dt. r→∞ r 0 0 To see this, we ﬁrst compute it for p(x) = xn. We have

1 Z ∞ 1 Z ∞ e−t/re−nt/r dβ(t) = e−(n+1)t/r dβ(t). r 0 r 0 Therefore, 1 1 Z ∞ C ω^-lim e−(n+1)t/r dβ(t) = n + 1 r→∞ r/(n + 1) 0 n + 1 by the dilation invariance of ωe. Thus, 1 Z ∞ Z ∞ ω^-lim e−t/re−nt/r dβ(t) = C e−t(e−t)n dt. r→∞ r 0 0

Since ωe is linear, the assertion follows for all p. Dixmier traces and some applications 1071

We choose sequences {pn}, {Pn} of polynomials such that for all x ∈ [0, 1]

−1 6 pn(x) 6 g(x) 6 Pn(x) 6 3 and such that pn and Pn converge almost everywhere to g(x). Then since ωe is positive, it preserves order: Z ∞ Z ∞ −t −t 1 −t/r −t/r C e pn(e ) dt = ω^-lim e pn(e ) dβ(t) 0 r→∞ r 0 Z ∞ Z ∞ 1 −t/r −t/r −t −t 6 ω^-lim e g(e ) dβ(t) 6 ··· 6 C e Pn(e ) dt. r→∞ r 0 0

R ∞ −t −t The Lebesgue dominated convergence theorem gives us that both 0 e pn(e ) dt R ∞ −t −t R ∞ −t −t and 0 e Pn(e ) dt converge to 0 e g(e ) dt as n → ∞. But a direct calcu- R ∞ −t −t lation yields 0 e g(e ) dt = 1 and Z ∞ e−t/rg(e−t/r) dβ(t) = β(r). 0 Hence, 1 Z ∞ β(r) C = ω^-lim e−t/rg(e−t/r) dβ(t) = ω^-lim . r→∞ r 0 r→∞ r This completes the proof of Theorem 10.2. We recall that for any τ-measurable operator T the distribution function of T is deﬁned by λt(T ) := τ(χ(t,∞)(|T |)), t > 0, where χ(t,∞)(|T |) is the spectral projection of |T | corresponding to the interval (t, ∞) (see [47]). By Proposition 2.2 in [47],

µs(T ) = inf{t > 0 : λt(T ) 6 s}, we infer that for any τ-measurable operator T the distribution function λ( · )(T ) coincides with the (classical) distribution function µ( · )(T ). From this formula and the fact that λ is right-continuous, we easily see that for t > 0 and s > 0

s > λt ⇐⇒ µs 6 t or, equivalently, s < λt ⇐⇒ µs > t. By Remark 3.3 in [47], this implies that

Z λt Z µs(T ) ds = µs(T ) ds = τ(|T |χ(t,∞)(|T |)), t > 0. (18) 0 [0,λt)

(1,∞) Lemma 10.4. For T ∈ L and C > kT k(1,∞),

λ 1 (T ) Ct log t t 6 for large t. 1072 A. L. Carey and F. A. Sukochev

Proof. Suppose not, that is, there exists a sequence tn ↑ ∞ such that λ 1 (T ) > tn Ctn log tn, so that for s 6 Ctn log tn we have µs(T ) > µCtn log tn (T ) > 1/tn. Then for suﬃciently large n

Z Ctn log tn 1 µs(T ) ds > · Ctn log tn = C log tn. 0 tn

We choose δ > 0 with C − δ > kT k(1,∞). Then for suﬃciently large n

C log tn = (C − δ) log tn + δ log tn > kT k(1,∞) log(Ctn) + kT k(1,∞) log log(tn + 1)

= kT k(1,∞) log(Ctn log(tn + 1)).

R t This contradicts the inequality 0 µs(T ) ds 6 kT k(1,∞) log(t + 1), which holds for any t > 0 by the deﬁnition of the norm in L (1,∞). Lemma 10.4 is proved. An assertion somewhat similar to Proposition 10.5 below was formulated in [36] but was supplied with an incorrect proof. We use a diﬀerent approach.

(1,∞) ∗ Proposition 10.5. For T ∈ L positive let ω be a state on L∞(R+) satisfying all the conditions of Corollary 4.8. For all C > 0

1 Z t 1 τω(T ) = ω-lim µs(T ) ds = ω-lim τ(T χ 1 (T )) ( t ,∞) t→∞ log(1 + t) 0 t→∞ log(1 + t) 1 Z Ct log t = ω-lim µs(T ) ds, t→∞ log(1 + t) 0 and if one of the ω-limits is a true limit, then so are the others. Proof. We ﬁrst note that

Z t Z λ 1 (T ) t µs(T ) ds 6 µs(T ) ds + 1, t > 0. 0 0

Indeed, the inequality above holds trivially if t λ 1 (T ). If t > λ 1 (T ), then 6 t t

Z t Z λ 1 (T ) Z t t µs(T ) ds = µs(T ) ds + µs(T ) ds. 0 0 λ 1 (T ) t

It follows from the inequality s > λ 1 (T ) that µs(T ) 1/t, therefore, t 6

Z t Z λ 1 (T ) Z λ 1 (T ) t 1 t µs(T ) ds µs(T ) ds + (t − λ 1 (T )) µs(T ) ds + 1. 6 t 6 0 0 t 0

Using this observation and the lemma above, we see that for C > kT k(1,∞) and any ﬁxed α > 1

α Z t Z λ 1 (T ) Z Ct log t Z t t µs(T ) ds 6 µs(T ) ds + 1 6 µs(T ) ds + 1 6 µs(T ) ds + 1 0 0 0 0 Dixmier traces and some applications 1073 for large t, and so

Z t Z λ 1 (T ) 1 1 t µs(T ) ds 6 µs(T ) ds + 1 log(1 + t) 0 log(1 + t) 0 1 Z Ct log t 6 µs(T ) ds + 1 log(1 + t) 0 α log(1 + tα) Z t 6 α µs(T ) ds + 1 log(1 + t) log(1 + t ) 0 for large t. Taking the ω-limit, we get that

Z λ 1 (T ) 1 t τω(T ) 6 ω-lim µs(T ) ds t→∞ log(1 + t) 0 1 Z Ct log t 6 ω-lim µs(T ) ds t→∞ log(1 + t) 0 α α Z t 6 ω-lim α µs(T ) ds = ατω(T ), t→∞ log(1 + t ) 0 where the last line uses Corollary 4.8 (4). Since this holds for all α > 1, we get the conclusion for ω-limits and C > kT k(1,∞) in view of (18). The assertion for an 0 arbitrary 0 < C 6 kT k(1,∞) follows immediately by noting that for C > kT k(1,∞) Z t Z Ct log t Z C0t log t µs(T ) ds 6 µs(T ) ds 6 µs(T ) ds 0 0 0 for large t. To see the last assertion of the proposition, suppose that

1 Z t lim µs(T ) ds = A; t→∞ log(1 + t) 0 then by the above argument we get that 1 1 A 6 lim inf τ(T χ( 1 ,∞)(T )) 6 lim sup τ(T χ( 1 ,∞)(T )) 6 αA t→∞ log(1 + t) t t→∞ log(1 + t) t for all α > 1, and hence 1 lim τ(T χ( 1 ,∞)(T )) = A. t→∞ log(1 + t) t On the other hand, if the limit 1 lim τ(T χ( 1 ,∞)(T )) t→∞ log(1 + t) t exists and equals B, then

1 Z t 1 Z t lim sup µs(T ) ds 6 B 6 α lim inf µs(T ) ds t→∞ log(1 + t) 0 t→∞ log(1 + t) 0 1074 A. L. Carey and F. A. Sukochev for all α > 1, so that 1 Z t lim µs(T ) ds = B t→∞ log(1 + t) 0 as well. The remaining assertions follow similarly. 10.2. The zeta function and the Dixmier trace. This subsection is motivated by Proposition IV.2.4 in [7]. We will describe several generalizations of this result to the von Neumann algebra setting. Our approach will be analogous to that in [7], though with diﬀerences. The zeta function of a positive T ∈ L (1,∞) is given by ζ(s) = τ(T s), while for s A ∈ N we set ζA(s) = τ(AT ). We are interested in the asymptotic behaviour of ζ(s) and ζA(s) as s → 1. It is now easy to see the importance of the discussion of singular traces, because by Lemma 10.1, Z ∞ Z ∞ Z ∞ s s s s s K K τ(T ) = µr(T ) dr = µr(T ) dr 6 s dr = 0 0 0 (1 + r) s − 1 for some K > 0 and all s > 1. From this it follows that {(s − 1)τ(T s): s > 1} s s is bounded. For bounded A we now have |(s − 1)τ(AT )| 6 kAk(s − 1)τ(T ), so s ∗ that (s − 1)τ(AT ) is also bounded, and hence for any ωe ∈ L∞(R) satisfying conditions (1), (2), and (3) of Theorem 4.7 the limit

1 1+ 1 ω^-lim τ(AT r ) (19) r→∞ r exists. 1 1 1+ r Here r 7→ r τ(AT ) is defined on all of R by extending it to be identically s zero for r < 1. One might like to think of (19) as ω^-lims→1(s − 1)τ(AT ), but this s of course does not (strictly speaking) make sense, whereas if lims→1(s − 1)τ(AT ) 1 1 1+ r exists, then it is limr→∞ r τ(AT ). (1,∞) In the following theorem we will take T ∈ L positive and such that kT k 6 1, with spectral resolution T = R λ dE(λ). We would like to integrate with respect to dτ(E(λ)); unfortunately, these scalars τ(E(λ)) are, in general, all infinite. To remedy this situation, we must instead integrate with respect to the increasing (negative) real-valued function NT (λ) = τ(E(λ) − 1) for λ > 0. Away from 0, the increments τ(4E(λ)) and 4NT (λ) are identical, of course. (1,∞) ∗ Theorem 10.6. For T ∈ L positive with kT k 6 1 and for an ωe ∈ L∞(R) satisfying all the conditions of Theorem 4.7, let ωe = ω ◦ L, where L is given in § 4 (before Definition 4.4). Then

1 1+ 1 τω(T ) = ω^-lim τ(T r ). r→∞ r 1+ 1 If limr→∞ τ(T r )/r exists, then

1 1+ 1 τω(T ) = lim τ(T r ) r→∞ r ∗ ∗ for an arbitrary dilation-invariant functional ω ∈ L∞(R+) . Dixmier traces and some applications 1075

1 1 1+ r ∗ Proof. By (19), we can apply to the function r τ(T ) the weak Karamata theorem. First we write Z 1 1+ 1 1+ 1 τ(T r ) = λ r dNT (λ). 0+ Setting λ = e−u, we thus obtain Z ∞ 1+ 1 − u τ(T r ) = e r dβ(u), 0

R 0 −v −v R u −v −v where β(u) = u e dNT (e ) = − 0 e dNT (e ). Since the change of variable λ = e−u is strictly decreasing, β is, in fact, non-negative and increasing. By the ∗ ∗ weak Karamata theorem applied to ωe ∈ L∞(R) ,

1 1+ 1 β(u) ω^-lim τ(T r ) = ω^-lim . r→∞ r u→∞ u With the substitution ρ = e−v we next get that β(u) 1 Z 1 ω^-lim = ω^-lim ρ dNT (ρ). (20) u→∞ u u→∞ u e−u Let f(u) = β(u)/u. We want to make the change of variable u = log t, or in other words, to consider f ◦ log = Lf. We use the discussion in § 4, which tells us ∗ ∗ that if we start with an M-invariant functional ω ∈ L∞(R+) , then the functional ωe = ω ◦ L is H-invariant, as required by the theorem. Then

1 1+ 1 β(u) ω^-lim τ(T r ) = ω^-lim = ω^-lim f(u) r→∞ r u→∞ u u→∞ 1 Z 1 = ω-lim Lf(t) = ω-lim λ dNT (λ). t→∞ t→∞ log t 1/t Now, by Proposition 10.5, 1 Z 1 1 ω-lim λ dNT (λ) = ω-lim τ(χ 1 (T )T ) = τω(T ). ( t ,1] t→∞ log t 1/t t→∞ log t This completes the proof of the ﬁrst part of the theorem. The proof of the second part is similar. Using the classical Karamata theorem (see Remark 10.3), we obtain the following analogue of (20): Z 1 1 1+r β(u) 1 lim τ(T ) = lim = lim ρ dNT (ρ). r→∞ r u→∞ u u→∞ u e−u Making the substitution u = log t on the right-hand side, we have 1 Z 1 1 Z 1 lim ρ dNT (ρ) = lim λ dNT (λ) = τω(T ), u→∞ u −u t→∞ log t 1 e t ∗ ∗ where in the last equality we need only dilation invariance of the state ω ∈ L∞(R+) and not the full list of conditions of Corollary 4.8. This completes the proof of Theorem 10.6. 1076 A. L. Carey and F. A. Sukochev

(1,∞) The map from the positive operators T ∈ L to R given by T 7→ τω(T ) can be extended by linearity to a C-valued functional on the whole of L (1,∞). Then the functional A 7→ τω(AT ) (21) is well deﬁned for A ∈ N and ﬁxed T ∈ L (1,∞). We intend to study the properties of (21). Part of the interest in this functional stems from the following result (see [61]) as well as from the use of the Dixmier trace in non-commutative geometry [7]. Lemma 10.7. Let T ∈ L (1,∞). Then the following assertions are valid. (i) For A ∈ N τω(AT ) = τω(TA).

(ii) Assume that D0 is an unbounded self-adjoint operator affiliated with N such 2 −1/2 (1,∞) that T = (1 + D0) ∈ L . If [Aj, |D0| ] is a bounded operator for Aj ∈ N , j = 1, 2, then τω(A1A2T ) = τω(A2A1T ). Proof. (i) This is proposition A.2 in [85]. The proof is elementary: we first show ∗ that τω(UTU ) = τω(T ), then use linearity to extend the equality to arbitrary T ∈ L (1,∞), then replace T by TU, and then use linearity again. (ii) We remark that [Aj, |D0| ] defining a bounded operator means that the Aj leave dom(|D0|) = dom(D0) invariant and that [Aj, |D0| ] is bounded on this domain (see [86], Proposition 3.2.55 and its proof for equivalent but seemingly weaker condi- 2 1/2 2 1/2 tions). Since |D0|−(1+D0) is bounded, [Aj, (1+D0) ] defines a bounded oper- −1 2 1/2 −1 ator whenever [Aj, |D0| ] does. Because T = (1+D0) and T : H → dom(T ), we see that the formal calculation

−1 −1 −1 [Aj,T ] = AjT − TAj = T (T Aj − AjT )T = T [T ,Aj]T makes sense as equalities of everywhere deﬁned bounded operators on H . That is, 2 1/2 (1,∞) 2 1 [Aj,T ] = T [(1 + D0) ,Aj]T ∈ (L ) ⊆ L . Then we have, by part (i),

τω(A1A2T ) = τω(A2A1T ) − τω([A1,T ]A2). Thus, 2 1/2 τω(A1A2T ) = τω(A2A1T ) − τω(T [(1 + D0) ,A1]TA2). Since the operator in the last term is trace class, we are done. We will consider spectral triples, which involves choosing a subalgebra of N on which (21) will deﬁne a trace. (1,∞) Theorem 10.8. Let A ∈ N , T > 0, T ∈ L . s (i) If lims→1+ (s − 1)τ(AT ) exists, then it is equal to τω(AT ), where ω is chosen as in the proof of Theorem 10.6. (ii) More generally, if the functionals ω and ωe are chosen as in the proof of Theorem 10.6, then 1 1+ 1 ω^-lim τ(AT r ) = τω(AT ). r→∞ r Dixmier traces and some applications 1077

Proof. For part (i) we ﬁrst assume that A is self-adjoint and write A = a+ − a−, ± where a are positive. If ωe is chosen as in the proof of Theorem 10.6, then

s 1 1+ 1 lim (s − 1)τ(AT ) = ω^-lim τ(AT r ) s→1+ r→∞ r 1 + 1+ 1 1 − 1+ 1 = ω^-lim τ(a T r ) − ω^-lim τ(a T r ) r→∞ r r→∞ r + − = τω(a T ) − τω(a T ) = τω(AT ).

Here the third equality uses a technical result in [29] (Proposition 3.6) and then Theorem 10.6. The reduction from the general case to the self-adjoint case now follows in a similar way. For part (ii), we assume that A is positive. From Lemma 10.7 (i), Theorem 10.6, and [29] (Proposition 3.6) we have

1/2 1/2 1 1/2 1/2 1+ 1 1 1+ 1 τω(AT ) = τω(A TA ) = ω^-lim τ((A TA ) r ) = ω^-lim τ(AT r ). r→∞ r r→∞ r For general A we reduce our arguments to the case when A is positive, as in the proof of part (i).

11. The heat semigroup formula This section is inspired by [7] and again is taken from [29]. We retain the −T −2 assumption T > 0 throughout and deﬁne e as the operator that is zero on ⊥ ker T and is deﬁned on ker T by the functional calculus. We remark that if T > 0 (p,∞) −tT −2 and T ∈ L for some p > 1, then e is a trace-class operator for all t > 0. Our aim in this section is to prove the following theorem.

(1,∞) Theorem 11.1. If A ∈ N , T > 0, and T ∈ L , then

−1 −λ−2T −2 ω-lim λ τ(Ae ) = Γ(3/2)τω(AT ) λ→∞

∗ ∗ for ω ∈ L∞(R+) satisfying the conditions of Corollary 4.8.

1 1 p+ r Let ζA p + r = τ(AT ). We observe that the limit

1 p 1 1 Γ ω^-lim ζA p + 2 2 r→∞ r r always exists. Hence, we can reduce the hard part of the proof of Theorem 11.1 to the following preliminary result.

(p,∞) Proposition 11.2. If A ∈ N , A > 0, T > 0, and T ∈ L , 1 6 p < ∞, then with ω and ωe chosen as in the proof of Theorem 10.6, 1 −T −2λ−2/p 1 p 1 1 ω-lim τ(Ae ) = Γ ω^-lim ζA p + . λ→∞ λ 2 2 r→∞ r r 1078 A. L. Carey and F. A. Sukochev

Proof. Using the Laplace transform, we have

∞ 1 Z −2 T s = ts/2−1e−tT dt. Γ(s/2) 0 Then Z ∞ s 1 s/2−1 −tT −2 ζA(s) = τ(AT ) = t τ(Ae ) dt. Γ(s/2) 0 Under the change of variable t = 1/λ2/p the preceding formula becomes Z ∞ p − s −1 −λ−2/pT −2 Γ(s/2)ζA(s) = λ p τ(Ae ) dλ. 2 0 R 1 R ∞ We split this integral into the two parts 0 and 1 and call the ﬁrst integral R(r), 1 where s = p + r . Then Z 1 Z ∞ − 1 −2 −λ−2/pT −2 p + 1 −1 −tT −2 R(r) = λ pr τ(Ae ) dλ = t 2 2r τ(Ae ) dt. 0 1

−2 2 −1 The integrand decays exponentially in t as t → ∞, because T > kT k 1, and hence −2 −2 t−1 −tT −T − 2 τ(Ae ) 6 τ(Ae e kT k ). Then we can conclude that R(r) is bounded uniformly with respect to r, and 1 µ therefore limr→∞ r R(r) = 0. For the other integral the change of variable λ = e gives Z ∞ Z ∞ − 1 −2 −λ−2/pT −2 − µ λ pr τ(Ae ) dλ = e pr dβ(µ), 1 0 where µ 2 Z − v −2 β(µ) = e−vτ(Ae−e p T ) dv. 0 Hence, we can now write Z ∞ p 1 1 1 − µ Γ p + ζA p + = e pr dβ(µ) + R(r). 2 2 r r 0 We now consider p 1 p p 1 p p 1 1 ω^-lim Γ + ζA p + = Γ ω^-lim ζA p + . 2 r→∞ r 2 2r r 2 2 r→∞ r r Then Z ∞ p p 1 1 1 − µ Γ ω^-lim ζA p + = p ω^-lim e pr dβ(µ) 2 2 r→∞ r r r→∞ pr 0 1 (remembering that the term r R(r) has limit zero as r → ∞). By dilation invariance and Theorem 10.2 we then have p p 1 1 β(µ) Γ ω^-lim ζA p + = p ω^-lim . (22) 2 2 r→∞ r r µ→∞ µ Dixmier traces and some applications 1079

Making the change of variable λ = ev in the expression for β(µ), we get that

eµ β(µ) 1 Z −2 −2/p = λ−2τ(Ae−T λ ) dλ. µ µ 1 With the substitution µ = log t the right-hand side becomes

Z t 1 −2 −T −2λ−2/p λ τ(Ae ) dλ = g1(t). log t 1 This is the Ces`aromean of

1 −2 −2/p g (λ) = τ(Ae−T λ ). 2 λ

∗ ∗ Then since we chose an ω ∈ L∞(R+) satisfying Corollary 4.8, we have ω(g1) = ω(g2). Recalling that we chose ωe to be related to ω as in Theorem 10.6, so that we can use (22), we obtain

1 −T −2λ−2/p 1 p 1 1 ω-lim τ(Ae ) = Γ ω^-lim ζA p + . λ→∞ λ 2 2 r→∞ r r This completes the proof of Proposition 11.2.

To prove the theorem let us consider ﬁrst the case when A is bounded and A > 0, and use Proposition 11.2 and Theorem 10.8 to get that

3 3 1 1+ 1 −1 −λ−2T −2 Γ τω(AT ) = Γ ω^-lim τ(AT r ) = ω-lim λ τ(Ae ). 2 2 r→∞ r λ→∞ Then for self-adjoint A we write A = a+ − a− with a± positive, so that

+ − Γ(3/2)τω(AT ) = Γ(3/2)(τω(a T ) − τω(a T )) −2 −2 −2 −2 = ω-lim λ−1τ(a+e−λ T ) − ω-lim λ−1τ(a−e−λ T ) λ→∞ λ→∞ −2 −2 = ω-lim λ−1τ(Ae−λ T ). λ→∞ We can extend this to arbitrary bounded A by a similar argument.

12. The case p > 1 The results of the previous sections have analogues for p > 1. Some arguments are simpler in this case due to the fact that the singular values satisfy the asymptotic 1/p (p,∞) estimate µs(T ) = O(1/s ) for T ∈ L if p > 1 and T > 0. Moreover, Z 1 p+ 1 p+1/r τ(T r ) = λ dNT (λ), 0 R where NT (λ) = τ(E(λ) − 1) for λ > 0, where T = λ dE(λ) is the spectral resolution for T . We now establish some L (p,∞)-versions of our previous results. 1080 A. L. Carey and F. A. Sukochev

(p,∞) Lemma 12.1. For T ∈ L and for ω and ωe as in the proof of Theorem 10.6,

p 1 p+ 1 pτω(T ) = ω^-lim τ(T r ). r→∞ r

Proof. Let λ = e−u/p, so that

Z ∞ 1 p+ 1 1 −u/(rp) τ(T r ) = p e dβ(u), r pr 0

R u −v −v/p where β(u) = 0 e dNT (e ). The dilation invariance gives us that

Z 1 1 p+ 1 1 −u/(pr) β(u) ω^-lim τ(T r ) = p ω^-lim e dβ(u) = p ω^-lim , r→∞ r r→∞ pr 0 u→∞ u by the weak∗ Karamata Theorem 10.2. Reasoning as in the proof of Theorem 10.6 and substituting λ = e−v/p and u = log t, we have

Z 1 β(u) 1 p ω^-lim = ω-lim λ dNT (λ) u→∞ u t→∞ log t t−1/p 1 p p p = ω-lim τ(χ( 1 ,∞)(T )T ) = τω(T ). t→∞ log t t

Lemma 12.1 is proved.

(p,∞) Corollary 12.2. Let T > 0, T ∈ L . Then

1 −T −2λ−2/p p 1 p+ 1 ω-lim τ(e ) = Γ 1 + ω^-lim τ(T r ). λ→∞ λ 2 r→∞ r

Proof. Combine Proposition 11.2 and Lemma 12.1.

The L (p,∞)-version of Theorem 10.8 and the following result of Connes are proved by following the same methods as for p = 1. One needs some straightforward extensions of various technical results as described in [29](§ 5).

(p,∞) Theorem 12.3. If A is bounded, T > 0, and T ∈ L with p > 1, then

−1 −λ−2/pT −2 p ω-lim λ τ(Ae ) = Γ(1 + p/2)τω(AT ). λ→∞

Finally, it is now a simple matter to obtain the following result.

(p,∞) Theorem 12.4. If A is bounded, T > 0, T ∈ L , and

lim (s − p)τ(AT s) s→p+

p exists, then it is equal to pτω(AT ). Dixmier traces and some applications 1081

13. Generalized Toeplitz operators and their index The index theory of generalized Toeplitz operators has a long history which can be traced in part from [87] and [88]. We will explain how the results established in earlier sections can be used to contribute to this theory. As before, P denotes the projection onto the non-negative spectral subspace of an unbounded self-adjoint operator D0 (with bounded inverse) and we are interested in the Breuer–Fredholm index of the operator P uP acting on P H . When N = (1,∞) L (H ) and D0 is part of an L -summable spectral triple, the most general such theorem in this type I situation is that in [85], where it is shown that

1 ∗ −1 ind(P uP ) = τω(u[D0, u ]|D0| ). 2 We now show that this formula holds when N is a general semiﬁnite von Neumann algebra. Remark. Of course, [85] concerns a much more general situation which would apply, (p,∞) for example, in the case when D0 is part of an L -summable spectral triple. The authors obtain a very general formula for the index of P uP in terms of sums of residues of zeta functions constructed from D0 under certain assumptions on the analytic properties of these zeta functions. Some, but not all, of the zeta functions in their formula are Dixmier traces. We will not attempt to describe this result here, since there is an excellent overview in the article [89]. The theorem in [85] is proved for the type I case and its generalization to the semiﬁnite von Neumann algebra setting is achieved in [25], [26]. We need a preliminary result ([29], Lemma 6.1).

Lemma 13.1. Let D0 be an unbounded self-adjoint operator aﬃliated with N such 2 −1/2 (1,∞) that (1 + D0) is in L . Let At and B be in N for t ∈ [0, 1], with At self-adjoint and t 7→ At continuous. Let Dt = D0 + At and let p be a real number with 1 < p < 4/3. Then the quantity

2 −p/2 2 −p/2 τ B(1 + D0) − B(1 + Dt ) is uniformly bounded with respect to t ∈ [0, 1] and p ∈ (1, 4/3). (1,∞) Theorem 13.2. Let (N , D0) be an L -summable Breuer–Fredholm module for the unital Banach ∗-algebra A , and let u ∈ A be a unitary element such that [D0, u] is bounded. Let P be the projection onto the non-negative spectral subspace of D0. Then with ω chosen as in Corollary 4.8,

∗ 1 ∗ 2 −p/2 ind(P uP ) = sf(D0, uD0u ) = lim (p − 1)τ(u[D0, u ](1 + D0 ) ) p→1+ 2

1 ∗ 2 −1/2 1 ∗ −1 = τω(u[D0, u ](1 + D ) ) = τω(u[D0, u ]|D0| ), 2 0 2 where the last equality holds only if D0 has a bounded inverse. Remark 13.3. (1) The equality

1 ∗ −1 ind(P uP ) = τω(u[D0, u ] |D0| ) (23) 2 1082 A. L. Carey and F. A. Sukochev proved above should be compared with Theorem IV.2.8 in [7]. In the case when N = L (H ), the right-hand side of (23) is a Hochschild 1-cocycle on A which is known to equal the Chern character of the L (1,∞)-summable Fredholm module (A , D0, H ). (2) Since any 1-summable module is clearly an L (1,∞)-summable module, the theorem implies that any unbounded 1-summable module must have a trivial pairing with K1(A ) and is therefore uninteresting from the homological point of view. u ∗ Proof. Let Dt = D0 + tu[D0, u ], t ∈ [0, 1]. Then by [24](§ 2.1, (2)) we have for each p > 1 that

Z 1 1 ∗ u 2 −p/2 ind(P uP ) = τ u[D0, u ](1 + (Dt ) ) dt. Cp/2 0 By Lemma 13.1, we now get that

∗ u 2 −p/2 2 −p/2 τ u[D0, u ][(1 + (Dt ) ) − (1 + D0 ) ] is uniformly bounded with respect to t and p for 1 < p < 4/3. Since Cep/2 → ∞ as p → 1+, we see that:

1 ∗ 2 −p/2 ind(P uP ) − τ(u[D0, u ](1 + D0 ) ) Cp/2 Z 1 1 ∗ u 2 −p/2 = τ(u[D0, u ](1 + (Dt ) ) ) dt Cp/2 0 Z 1 1 ∗ 2 −p/2 − τ(u[D0, u ](1 + D0 ) ) dt Cp/2 0 Z 1 1 ∗ u 2 −p/2 2 −p/2 6 |τ(u[D0, u ][(1 + (Dt ) ) − (1 + D0 ) ])| dt Cp/2 0

6 const/Cp/2 → 0. It is now elementary that as p → 1+

2 Z 1 p Z ∞ 1 p = dx ∼ √ dx = Cep/2. p − 1 |x| 2 |x|>1 −∞ 1 + x This ends the proof of the ﬁrst equality, while the second equality follows from Theorem 10.8 (i). The third equality follows from the equality q q q 2−1 −1 2−1 −1 2 −1 1 + D0 − |D0| = 1 + D0 |D0| 1 + D0 + |D0| .

14. Non-smooth foliations and pseudodifferential operators In this section we give a further application of the earlier results, following [22] and [36]. The main aim of Prinzis’ thesis [36] was to establish a Wodzicki residue formula for the Dixmier trace of certain pseudodifferential operators associated with Dixmier traces and some applications 1083 actions of Rn on a compact space X. This was greatly generalized in [22], but to cover that paper in detail would require an extensive discussion of foliations, so we restrict ourselves here to a synopsis of the simple case in [36]. The set-up is the ‘group-measure space’ construction of Murray and von Neu- mann. Thus, X is a compact space equipped with a probability measure ν and a continuous free minimal ergodic action α of Rn on X leaving ν invariant. We write the action as x → t.x for x ∈ X and t ∈ Rn. Then the crossed prod- n uct L∞(X, ν) ×α R is a type II factor contained in the bounded operators on 2 n 2 1 L (R ,L (X, ν)). We describe the construction. For f ∈ L (R,L∞(X, ν)) ⊂ n 2 n 2 L∞(X, ν) ×α R the action of f on a vector ξ in L (R ,L (X, ν)) is defined by twisted left convolution as follows: Z −1 (π(f)ξ)(s) = αs (f(t))ξ(s − t) dt. e n R Here f(t) is a function on X acting as a multiplication operator on L2(X, ν). The twisted convolution algebra

1 n 2 n 2 L (R ,L∞(X, ν)) ∩ L (R ,L (X, ν)) is a dense subspace of L2(Rn,L2(X, ν)) and there is a canonical faithful normal semifinite trace Tr on the von Neumann algebra that it generates. This von Neu- mann algebra is n 00 N = (πe(L∞(X, ν) ×α R )) . n 2 n 2 For functions f, g : R → L∞(X) which are in L (R ,L (X, ν)) and whose twisted 2 n 2 left convolutions πe(f), πe(g) define bounded operators on L (R ,L (X, ν)), this trace is given by: Z Z Tr(π(f)∗π(g)) = f(t, x)g(t, x)∗ dν(x) dt, e e n R X where we think of f and g as functions on Rn × X. We identify L2(Rn) with L2(Rn) ⊗ 1 ⊂ L2(R,L2(X, ν)). Then any scalar func- n 2 tion f on R that is the Fourier transform f = gb of a bounded L -function g will 2 2 be in L (R,L (X, ν)), and πe(f) will be a bounded operator. Pseudodifferential operators are defined in terms of their symbols. A smooth symbol of order m is a function a: X × Rn → C such that for each x ∈ X the function ax defined by ax(t, ξ) = a(t.x, ξ) satisfies β γ −m+|β| n n n (1) sup{|∂ξ ∂t ax(t, ξ)(1 + |ξ|) | :(t, ξ) ∈ R × R , β, γ ∈ N , |β| + |γ| 6 M} < ∞ for all M ∈ N, n ∞ (2) ξ 7→ ax(0, ξ) is a smooth function on R into the space C (X) of continuous functions f on X such that the function t 7→ (x 7→ f(t.x)) is smooth on Rn. ∞ n Each symbol a defines a pseudodifferential operator Op(a) on C(X) ⊗ Cc (R ) by Z 1 itξ ∞ n Op(a)f(x, t) = n e a(t.x, ξ)fb(x, ξ) dξ, f ∈ C(X) ⊗ Cc (R ), (2π) n R 1084 A. L. Carey and F. A. Sukochev

∞ n where Cc (R ) is the space of smooth functions with compact support. The principal symbol of a pseudodiﬀerential operator A on X is the limit

a(x, λξ) n σm(A)(x, ξ) = lim , (x, ξ) ∈ X × R \{0}, λ→∞ λm if it exists. We say that A is elliptic if its symbol a is such that ax is elliptic (that is, invertible for all suﬃciently large ξ) for all x ∈ X. Prinzis studies invertible positive elliptic pseudodiﬀerential operators A with a principal symbol. Henceforth we will consider only such operators. The zeta function of such an operator is ζ(z) = τ(Az), and this exists, because Az is a trace-class operator [36] for Re z < −n/m. Prinzis shows that n 1 Z n − m lim x + ζ(x) = − n σm(A)(x, ξ) dν(x) dξ (24) n − n−1 x→− m m (2π) m X×S

− n (1,∞) and that A m ∈ L (N , τ). Let us now observe that (24) combined with Theorem 12.4 implies that

n 1 Z n − m − m τω(A ) = n σm(A)(x, ξ) dν(x) dξ. (2π) n X×Sn−1 In other words, we have a type II Wodzicki residue for evaluating the Dixmier trace of these pseudodiﬀerential operators. The paper [22] considers the case of measured foliations in the sense of [90]. Thus, the leaves of the foliation are no longer given by an action of Rn as above, but are more general submanifolds. The main result of [22] on this topic is a formula for the Dixmier trace of certain pseudodiﬀerential operators in terms of a local residue, where the latter is a generalization of the Wodzicki residue to the foliation setting.

15. The algebra of almost periodic functions This section is adapted from [79].

15.1. Almost periodic pseudodifferential operators. We review Shubin’s study [91] of the index theory of differential operators with almost periodic coefficients, which develops ideas in [87]. Let us recall that a trigonometric function ihx,ξi is a finite linear combination of exponential functions eξ : x 7→ e . The space n ∗ n Trig(R ) of trigonometric functions is a ∗-subalgebra of the C -algebra Cb(R ) of bounded continuous functions. The uniform closure of Trig(Rn) is called the algebra of almost periodic functions and is denoted by A P (Rn). This C∗-algebra is n isomorphic to the algebra of continuous functions on RB, the Bohr compactifica- n n n tion of R . The addition in R extends to RB, which is a compact Abelian group n n containing R as a dense subgroup. The normalized Haar measure αB on RB is n such that the family (eξ)ξ∈R is orthonormal. The measure αB is given for any almost periodic function f on Rn by 1 Z αB(f) := lim n f(x) dx. T →+∞ (2T ) (−T,T )n Dixmier traces and some applications 1085

2 n n Using αB, one deﬁnes the Hilbert space completion L (RB) of Trig(R ). This n Hilbert space has an orthonormal basis given by (eξ)ξ∈R . The Fourier transform 2 n 2 n FB : ` (Rd ) → L (RB) is determined by

FB(δξ) = eξ, where δξ(η) = δξ,η is the Kronecker symbol.

We denote by F the usual Fourier transform on the Abelian group Rn with its usual Lebesgue measure. n n The action of R on RB by translations yields a topological dynamical system whose naturally associated von Neumann algebra is the crossed product n n L∞(RB) × R . It is more convenient for applications to consider the commutant N of this von Neumann algebra. It is also a crossed product, namely the von n n Neumann algebra L∞(R ) × Rd . Then N is a type II∞ factor with a faithful normal semiﬁnite trace τ. It can be described as the set of Borel essentially bounded 2 n n families (A ) n of bounded operators on L ( ) which are -equivariant, that µ µ∈RB R R is, such that n Aµ = σµ(A0) = T−µA0Tµ ∀ µ ∈ R .

Here and below we denote by σµ conjugation of any operator with the translation Tµ: σµ(B) = T−µBTµ. If we denote by Mϕ the operator of multiplication by a bounded function ϕ, then examples of such families are given for any λ by the families (σ (M )) n . µ eλ µ∈RB We choose the Fourier transform 1 Z f(ζ) = eixζ f(x) dx. F n/2 (2π) n R Then the von Neumann algebra N can be deﬁned [87] as the completion on the 2 n 2 n Hilbert space H = L (R ) ⊗ L (RB) of the set of operators {Meλ ⊗ Meλ ,Tλ ⊗ 1}, where λ ranges over Rn. There is a natural way to embed the C∗-algebra A P (Rn) in N by setting

π(f) := (σ (M )) n . µ f µ∈RB

This family then belongs to N and π is clearly faithful. Viewed as an operator on H , π(f) is given by π(f)(g)(x, µ) = f(x + µ)g(x, µ). If B = (Bµ)µ is a positive element of N , then we deﬁne the expectation E(B) as the Haar integral: Z E(B) := Bµ dαB(µ). n RB

Since the family B and the measure αB are translation invariant, the operator E(B) clearly commutes with translations on L2(Rn) and is therefore given by a Fourier ∞ n multiplier Mf(ϕB), where ϕB is a positive element of L (R ). We recall that the Fourier multiplier Mf(ϕB) is conjugation of the multiplication operator Mϕ by −1 the Fourier transform, that is, Mf(ϕB) = F MϕF . When the function ϕ is 1086 A. L. Carey and F. A. Sukochev in the Schwartz space, the operator Mf(ϕB) is convolution with the Schwartz function (2π)−n/2F −1ϕ. Hence, the expectation E takes values in the von Neumann algebra Mf(L∞(Rn)), that is, ∞ n E : N → Mf(L (R )). Using the usual Lebesgue integral on Rn and the normalization of Coburn et al. in [87], we now introduce the following deﬁnition of the trace τ: Z τ(B) = ϕB(ζ) dζ. n R Lemma 15.1 [87]. The map τ is, up to constant, the unique positive normal faithful semiﬁnite trace on N .

We consider the trace on N , evaluated on an operator of the form MaK, where a is almost periodic and K is the convolution operator on L2(Rn) arising from multiplication by an L1-function k after applying the Fourier transform. We have 1 Z Z τ(MaK) = lim n a(x) dx k(ζ) dζ. T →+∞ (2T ) n n (−T,T ) R More generally, any pseudodifferential operator A of non-positive order m acting 2 n N n in L (R , C ) with almost periodic coefficients can be viewed as a family over RB of pseudodifferential operators on Rn. To do this, we first take the symbol a of A; then the operator σµ(A) is the pseudodifferential operator with almost periodic coefficients whose symbol is (x, ξ) 7→ a(x + µ, ξ).

0 When m 6 0, we obtain in this way an element of N . We denote by ΨAP the algebra of pseudodifferential operators with almost periodic coefficients and with non-positive order. If the order m of A is greater than 0, then the operator A] given by the family (σ (A)) n is affiliated with . If the order m of A is less µ µ∈RB N than −n, then the bounded operator A] is a trace-class operator with respect to the trace τ on the von Neumann algebra N ⊗ MN (C) ([92], Proposition 3.3), and we have Z ] 1 τ(A ) = lim n Tr(a(x, ζ)) dx dζ. T →+∞ (2T ) n n (−T,+T ) ×R Indeed, the expectation E(A]) is a pseudodifferential operator on Rn with symbol (denoted by E(a) and independent of x) given by 1 Z E(a)(ζ) = lim n a(x, ζ) dx. T →+∞ (2T ) (−T,+T )n

Hence, the operator E(A]) is precisely the Fourier multiplier Mf(E(a)), and thus Z τ(A]) = Tr(E(a)(ζ)) dζ. n R ∞ Let ΨAP be the space of one-step polyhomogeneous classical pseudodiﬀerential operators on Rn with almost periodic coeﬃcients. Dixmier traces and some applications 1087

Theorem 15.2. Let A be a (scalar) pseudodiﬀerential operator with almost periodic coeﬃcients on Rn, assume that the order m of A is at most −n, and denote ] by a−n the −n homogeneous part of the symbol a. Then the operator A belongs to 1,∞ ] ] the Dixmier ideal L (N , τ). Moreover, the Dixmier trace τω(A ) of A associated with a limiting process ω does not depend on ω and is given by the formula Z ] 1 τω(A ) = a−n(x, ζ) dαB(x) dζ. n n n−1 RB ×S

Proof. As usual, ∆ is the Laplace operator on Rn. The operator A(1 + ∆)n/2 is then a pseudodiﬀerential operator with almost periodic coeﬃcients and non-positive order. Hence, the operator [A(1 + ∆)n/2]] = A](1 + ∆])n/2 belongs to the von Neumann algebra N . Further, the operator (1 + ∆])−n/2 is the Fourier multiplier 2 −n/2 determined by the function ζ 7→ (1+ζ ) . Hence if Eλ is the spectral projection of the operator (1 + ∆)−n/2 corresponding to the interval (0, λ), then the operator −n/2 1 − Eλ = χ(λ,+∞)(1 + ∆) is a Fourier multiplier as well. Suppose that it is 2 −n/2 determined by the function ζ 7→ fλ((ζ + 1) ). It follows that the trace τ of the operator 1 − Eλ is given by

Z 1 f dζ. λ 2 n/2 n (ζ + 1) R It is easy to compute this integral and to show that it is proportional to 1/λ. Then the inﬁmum of those λ for which τ(1 − Eλ) 6 t is precisely proportional to 1/t. Therefore, the operator (1 + ∆])−n/2, and hence A, belongs to the Dixmier ideal L1,∞(N , τ). In order to compute the Dixmier trace of the operator A, we use Theorem 10.1 in [91] to deduce that the spectral τ-density NA(λ) of A has the asymptotic expansion

γ (A) N (λ) = 0 (1 + o(1)), λ → +∞, A λ where γ0(A) is given by

1 Z γ0(A) = a−n(x, ζ) dαB(x) dζ. n n n−1 RB ×S

Now if A is positive, then by [22] (Proposition 1),

τω(A) = lim λNA(λ) = γ0(A). λ→+∞

This proves the theorem for positive A. Since the principal symbol map is a homomorphism, we deduce the result for general A. This completes the proof of Theo- rem 15.2. The normalization we have chosen for the trace in the von Neumann algebra setting of this section eliminates the factor 1/(2π)n which occurs with the Wodzicki residue in the type I theory. 1088 A. L. Carey and F. A. Sukochev

15.2. Almost periodic spectral triple. We denote by A a ∗-subalgebra of the algebra A P ∞(Rn) of smooth almost periodic functions on Rn. We take the Hilbert space on which the algebra N acts to be B2(Rn)⊗L2(Rn), where B2(Rn) is the Hilbert space of almost periodic functions on Rn, with the norm and inner product given by the restriction to A of the Haar trace on A P ∞(Rn) (note that 2 n ∼ 2 n B (R ) = ` (Rd )). This type II∞ von Neumann algebra is endowed with a faithful normal semifinite trace τ. (The explicit formula for τ is as given in the previous subsection.) n The usual Dirac operator on R is denoted by D0. Hence if S is the spin n n representation of R , then D0 acts on smooth S -valued functions on R . The n operator D0 is Z -periodic and is affiliated with the von Neumann algebra NS = N ⊗ End(S ). This latter is also a type II∞ von Neumann algebra with the trace τ ⊗ Tr. More generally, for any N > 1 we denote by NS ,N the von Neumann algebra N ⊗ End(S ⊗ CN ) with the trace τ ⊗ Tr. The algebra A and its closure are faithfully represented as ∗-subalgebras of the von Neumann algebra NS . In the same way, the algebra A ⊗MN (C) can be viewed ] as a *-subalgebra of NS ,N . More precisely, if a ∈ A , then the operator a defined by ] 2 n 2 n (a f)(x, y) := a(x + y)f(x, y) ∀ f ∈ B (R ) ⊗ L (R ) ] belongs to NS,N . The operator a is just the operator associated with the 0th-order differential operator corresponding to multiplication by a. The same formula enables us to represent A in NS . ] Proposition 15.3. The triple (A , NS , D0) is a semifinite spectral triple of finite dimension equal to n.

Proof. Note that the algebra A is unital. The differential operator D0 is known n ] to be elliptic, periodic, and self-adjoint on R . Therefore, the operator D0 is affiliated with the von Neumann algebra NS and is self-adjoint as a densely defined unbounded operator acting in the Hilbert space B2(Rn) ⊗ L2(Rn); more- 2 over, D0 = ∆⊗Id, where ∆ is the usual Laplacian. For any smooth almost periodic n function f on R , the commutator [D0, f] is a 0th-order almost periodic differential ] operator, and hence [D0, f] belongs to the von Neumann algebra N . On the other hand, the pseudodifferential operator T = (∆+I)−1/2 is the Fourier multiplier associated with the function 1 k 7→ . (kkk2 + 1)1/2

Therefore, its singular numbers µt(T ) can be computed explicitly as in the proof of Theorem 15.2 and thereby shown to be proportional to t−1/n.

16. Lesch’s index theorem In this section we describe (following [29]) the proof of an index theorem due to Lesch [88] (see also [74]) for Toeplitz operators with non-commutative symbols that is based on the zeta function approach to the Dixmier trace. We begin with ∗ a unital C -algebra A with a faithful ﬁnite trace τA satisfying τA (1) = 1 and a continuous action α of R on A leaving τA invariant. Dixmier traces and some applications 1089

We let HτA denote the Hilbert space completion of A in the inner product (a | b) = τ(b∗a). Then A is a Hilbert algebra and the left regular representation of A on itself extends by continuity to a representation a 7→ πτA (a) of A on

HτA [93]. In what follows, we will drop the notation πτA and just denote the action of A on HτA by juxtaposition. ∗ We now look at the induced representation πe of the crossed product C -algebra × on L2( ,H ). That is, π is the representation π × λ obtained from the A α R R τA e covariant pair (π, λ) of representations of the system (A , R, α) deﬁned for a ∈ A , 2 t, s ∈ R, and ξ ∈ L (R,HτA ) by

−1 (π(a)ξ)(s) = αs (a)ξ(s) and

λt(ξ)(s) = ξ(s − t).

1 Then for a function x ∈ L (R, A ) ⊂ A ×α R the action of πe(x) on a vector ξ in 2 L (R,HτA ) is deﬁned as follows: Z ∞ −1 (πe(x)ξ)(s) = αs (x(t))ξ(s − t) dt. −∞

1 2 Now the twisted convolution algebra L (R, A ) ∩ L (R,Hτ ) is a dense subspace 2 of L (R,Hτ ) and also a Hilbert algebra with respect to the given inner product. As such, there is a canonical faithful normal semiﬁnite trace τ on the von Neumann algebra 00 N = (πe(A ×α R)) . 2 For functions x, y : R → A ⊂ HτA in L (R,HτA ) whose twisted left convolutions π(x) and π(y) deﬁne bounded operators on L2( ,H ), this trace is given by e e R τA Z ∞ ∗ ∗ τ(πe(y) πe(x)) =: hx | yi = τA (x(t)y(t) ) dt. −∞

In particular, if we make the identiﬁcation L2( ) = L2( ) ⊗ χ ⊂ L2( ,H ), R R A R τA then any scalar function x on R that is the Fourier transform x = fb of a bounded L2-function f will have the properties that x ∈ L2( ,H ) and π(x) is R τA e a bounded operator. For such scalar functions x, the operator πe(x) is just the usual convolution with the function x and is usually denoted by λ(x), since it is the integrated form of the representation λ. The next lemma follows easily from these considerations. Lemma 16.1. With the hypotheses and notation discussed above: 2 (i) if h ∈ L (R), λ(h) is bounded, and a ∈ A , and if f : R → Hτ is deﬁned by f(t) = ah(t), then f ∈ L2( ,H ) and π(f) = π(a)λ(h) is bounded; R τA e 1 ∞ (ii) if g ∈ L (R) ∩ L (R) and a ∈ A , then π(a)λ(gb) is a trace-class operator in N , and Z ∞ Tr(π(a)λ(gb)) = τA (a) g(t) dt. −∞ 1090 A. L. Carey and F. A. Sukochev

2 Proof. To see part (i), let ξ ∈ Cc(R,HτA ) ⊆ L (R,HτA ). Then Z ∞ Z ∞ −1 −1 (πe(f)ξ)(s) = αs (f(t))ξ(s − t) dt = αs (a)h(t)ξ(s − t) dt −∞ −∞ Z ∞ −1 −1 = αs (a) h(t)ξ(s − t) dt = αs (a)(λ(h)ξ)(s) = (π(a)λ(h)ξ)(s). −∞

To see part (ii), we can assume that g is non-negative and a is self-adjoint. Then g1/2 ∈ L2 ∩ L∞, and so λ(gd1/2) is bounded. Now we have

1/2 1/2 π(a)λ(gb) = π(a)λ(gd )π(χA )λ(gd ).

Then π(a)λ(g1/2) = π(x), where x(t) = ag1/2(t), and π(χ )λ(g1/2) = π(y), where d e d A d e y(t) = χ g1/2(t). Thus, π(x) and π(y) are in and π(a)λ(g) = π(x)π(y). A d e e Nsa b e e Consequently, Z ∞ τ(π(a)λ(gb)) = τ(πe(x)πe(y)) = τ(x(t)y(t)) dt −∞ Z ∞ Z ∞ 1/2 2 = τA (a) gd (t) dt = τA (a) g(s) ds. −∞ −∞ This completes the proof of Lemma 16.1. By construction, N is a semifinite von Neumann algebra with a faithful normal semifinite trace τ and a faithful representation π : A → N [93]. For each t ∈ R, λt is a unitary operator in U(N ). In fact, the one-parameter unitary group {λt : itD t ∈ R} can be written as λt = e , where D is the unbounded self-adjoint operator 1 d D = , 2πi ds which is affiliated with N . In the Fourier transform picture of the previous proposition (that is, the spectral picture for D), D becomes multiplication by the independent variable, and so f(D) becomes pointwise multiplication by the function f. That is, πe(fb) = λ(fb) = f(D). Consequently, if f is a bounded L1-function, then Z ∞ τ(f(D)) = f(t) dt. −∞ By this discussion and the previous lemma, we have the following result.

Lemma 16.2. If f ∈ L1(R) ∩ L∞(R) and a ∈ A , then π(a)f(D) is a trace-class operator in N and Z ∞ τ(π(a)f(D)) = τ(a) f(t) dt. −∞ Dixmier traces and some applications 1091

We let δ be the densely defined (unbounded) ∗-derivation on A which is the infinitesimal generator of the representation α: R → Aut(A ), and we let δb be the unbounded ∗-derivation on N which is the infinitesimal generator of the representation Ad ◦λ: R → Aut(N ) (here Ad(λt) denotes the conjugation of the form ∗ λt · λt). Now if a ∈ dom(δ), then clearly π(a) ∈ dom(δb) and π(δ(a)) = δb(π(a)). By Proposition 3.2.55 in [86] (and its proof), π(δ(a)) leaves the domain of D invariant and π(δ(a)) = 2πi[D, π(a)]. We are now in a position to state and prove Lesch’s index theorem. ∗ Theorem 16.3 [29]. Let τA be a faithful finite trace on the unital C -algebra A , and let it be invariant under an action α of R. Let N be the semifinite von Neu- 00 mann algebra (πe(A ×αR)) , and let D be the infinitesimal generator of the canonical representation λ of R in U(N ). Then the representation π : A → N defines an L (1,∞)-summable Breuer–Fredholm module (N , D) for A . Moreover, if P is the non-negative spectral projection for D and u ∈ U(A ) is also in the domain of δ, then Tu := P π(u)P is a Breuer–Fredholm operator in P N P and 1 ind(T ) = τ(uδ(u∗)). u 2πi Proof. It is easy to see that D satisfies the condition (1 + D 2)−1/2 ∈ L (1,∞). By the previous discussion, for any a ∈ dom(δ) we have π(δ(a)) = 2πi[D, π(a)]. Since the domain of δ is dense in A , we see that π defines an L (1,∞)-summable Breuer–Fredholm module for A . By Theorem 13.2,

1 ∗ 2 −p/2 ind(Tu) = lim (p − 1) Tr π(u)[D, π(u )](1 + D ) ), p→1+ 2 and hence

1 1 ∗ 2 −p/2 ind(Tu) = lim (p − 1) Tr(π(uδ(u ))(1 + D ) ) p→1+ 2 2πi 1 1 Z ∞ = lim (p − 1) τ(uδ(u∗)) (1 + t2)−p/2 dt p→1+ 2 2πi −∞ 1 ∗ 1 1 ∗ = lim τ(uδ(u )) (p − 1)Cp/2 = τ(uδ(u )). p→1+ 2πi 2 2πi

17. The Hochschild class of the Chern character The theorem we discuss in this section, for N = L (H ) and 1 < p < ∞ (p integral), was proved in lectures by Alain Connes at the Collègede France in 1990. A version of this argument appeared in [8]. The extension of the argument to general semifinite von Neumann algebras, with the additional hypothesis that the unbounded self-adjoint operator D (which will form part of a spectral triple) has bounded inverse, was presented by Benameur and Fack in [22]. A simpler strategy using the Connes–Moscovici pseudodifferential calculus [85] was communicated to us by Nigel Higson. In conjunction with the results in [29], Higson’s argument 1092 A. L. Carey and F. A. Sukochev appears to generalize to the semifinite case, however, we will not describe the details here, focusing instead on another approach. In [25] the Connes and Benameur–Fack theorems were extended, first for p = 1, and then the hypothesis that D has bounded inverse was removed in the type II∞ case. This is crucial, due to the ‘zero-in-the-spectrum’ phenomenon for D. That is, for N of type II zero is generally in the point and/or continuous spectrum [94], and hence we are not dealing with just the simple problem that arises in the type I case posed by a finite-dimensional kernel. One feature of the approach in [25] is that the strategy of the proof is the same for all p > 1, and is independent of the type of the von Neumann algebra N . We explain the general semifinite version of the type I result in [7] (IV.2.γ), which computes the Hochschild class of the Chern character of a (p, ∞)-summable spectral triple. Theorem 17.1. Let (A , H , D) be a (p, ∞)-summable spectral triple of class QCk with integral p > 1 and k = max{2, p − 2}. Then: 1) a Hochschild cocycle on A is defined by

2 −p/2 ϕω(a0, . . . , ap) = λpτω(Γa0[D, a1] ··· [D, ap](1 + D ) );

2) for all Hochschild p-cycles c ∈ Cp(A )(that is, bc = 0),

hϕω, ci = hChFD , ci,

where ChFD is the Chern character in the cyclic cohomology of the pre-Fredholm 2 −1/2 module over A with FD = D(1 + D ) . ∗ Remark. Here τω is the Dixmier trace associated with any state ω ∈ D(R+). The two most important corollaries of Theorem 17.1 are the following. P i i i Corollary 17.2. With (A , H , D) as in Theorem 17.1, if c = i a0 ⊗a1 ⊗· · ·⊗ap is a Hochschild p-cycle, then the operator

X i i i 2 −p/2 Γ a0[D, a1] ··· [D, ap](1 + D ) i is (Dixmier-) measurable. This corollary is relevant to the axioms of non-commutative spin geometry, since it tells us that when we use the non-commutative integration theory provided by the Dixmier trace, we do not need to worry which functional ω we use to deﬁne the integration when we apply it to Hochschild cycles.

Corollary 17.3. With (A , H , D) as in Theorem 17.1, if ChFD pairs non-trivially with HHp(A ), then 2 −p/2 τω((1 + D ) ) 6= 0. Remark. The hypothesis of the Corollary is that there exists some Hochschild p-cycle such that hI ChFD , ci 6= 0, where I is the map deﬁned by Connes ([7], § III.1.γ, pp. 199–200). Computing this pairing using Theorem 17.1, we see that (1+D 2)−p/2 cannot have zero Dixmier trace for any choice of Dixmier functional ω. Dixmier traces and some applications 1093

2 −p/2 P i i For if (1 + D ) did have vanishing Dixmier trace, and c = i a0 ⊗ · · · ⊗ ap is any Hochschild cycle, then

X i i i 2 −p/2 |hI ChF , ci| = τω Γa [D, a ] ··· [D, a ](1 + D ) D 0 1 p i X i i i 2 −p/2 6 kΓa0[D, a1] ··· [D, ap] kτω (1 + D ) = 0. i Hence if the pairing is non-trivial, then the Dixmier trace cannot vanish on (1 + D 2)−p/2. There are subtle points in the proof of Theorem 17.1. One ﬁrst assumes that the 2 triple (A, H , D) has D invertible, replacing (A, H , D) by (A, H , Dm) with Dm = D m m −D if necessary, and then verifying that the ϕω in Theorem 17.1 does not depend on this replacement. One must also verify that the functional ϕω is indeed a Hochschild cocycle, but this is fairly simple.

Lemma 17.4. Let p > 1 and suppose that (A , H , D) is a (p, ∞)-summable spectral triple of class QC1. Then the multilinear functional

2 −p/2 ϕω(a0, . . . , ap) = λpτω(Γa0[D, a1] ··· [D, ap](1 + D ) ) is a Hochschild cocycle. Proof. By Lemma 3 of [25] and the trace property of the Dixmier trace, we have

p−1 2 −p/2 (bϕω)(a0, . . . , ap) = (−1) λpτω(Γa0[D, a1] ··· [D, ap−1]ap(1 + D ) ) p−1 2 −p/2 − (−1) λpτω(Γa0[D, a1] ··· [D, ap−1](1 + D ) ap). Since (A , H , D) is QC1,

p−1 2 −p/2 X 2 −(p−k)/2 2 1/2 2 −(1+k)/2 [(1 + D ) , ap] = − (1 + D ) [(1 + D ) , ap](1 + D ) k=0

2 −p/2 2 −p/2 and this is a trace-class operator. Thus, ap(1 + D ) = (1 + D ) ap modulo trace-class operators, and so the two terms above cancel.

18. Lidskii-type formula for Dixmier traces A semifinite analogue of the classical Lidskii theorem stated in terms of the (so-called) Brown spectral measure µT of an operator T ∈ N asserts [95] that Z τ(T ) = λ dµT (λ). σ(T )\{0} In the case when N = L (H ) and τ is the standard trace Tr, the equality above reduces to the classical case asserting that the trace Tr(T ) of an arbitrary trace-class P operator T is given by the sum λn(T ), where {λn(T )}n 1 is the sequence n>1 > of eigenvalues of T , arranged in decreasing order of absolute value and counting P multiplicities. We note that in the case T 0 the equality Tr(T ) = λn(T ) > n>1 1094 A. L. Carey and F. A. Sukochev follows immediately from the spectral theorem for compact operators. If T ∗ = T , then, again by the spectral theorem, we can select an orthonormal basis in H consisting of eigenvectors of T and still deduce Lidskii’s theorem without any difficulty. Here it is worth observing that the assumption that T ∗ = T is in the ideal L 1(H ) of all trace-class operators on H implies the absolute convergence of P the series |λn(T )|. The latter fact guarantees the convergence of the series P n>1 λn(T ) whatever the ordering of the set of eigenvalues of T (in particular, for n>1 the decreasing order of the absolute values). The core difference of this situation from the setting of Dixmier traces consists P in the fact that the series |λn(T )| diverges for any normal operator T ∈ n>1 L (1,∞)(H ) \ L 1(H ). Therefore, even though for a given T = T ∗ ∈ L (1,∞)(H ) we can define τω(T ) as the difference τω(T+)−τω(T−), where each of these numbers is computed according to the definition

N 1 X ω-lim λn(T±), N→∞ log(1 + N) n=1 it is by no means clear that

N 1 X τω(T ) = ω-lim λn(T ) N→∞ log(1 + N) n=1 for the special enumeration of the set {λ (T )} given by the decreasing order n n>1 of the absolute values |λn(T )|, or for that matter for any enumeration of this set. This difficulty becomes even more pronounced in the case of a general semifinite von Neumann algebra. (1,∞) The restriction µt(T ) 6 C/t, t > 1, imposed on an operator T ∈ L (N , τ) in the theorem below, implies that T belongs to the ideal N(ψ1)(N , τ) (see the definition of ψ1 in § 2.1 and the definition of the rearrangement-invariant ideal N(ψ) ∗ in § 2.2). Everywhere below, we assume that ω ∈ CD(R+) (see § 5.2). When we apply ω to a sequence x ∈ `∞, we identify x with its image i(x) in L∞(R+) (see Remark 5.3). Theorem 18.1 [21]. (i) If (N , τ) is a semifinite von Neumann algebra and T ∈ (1,∞) L (N , τ) satisfies the condition µt(T ) 6 C/t, t > 1, for some C > 0, then 1 Z τω(T ) = ω-lim λ dµT (λ). t→∞ log(1 + t) 1 λ/∈ t G

(ii) Let T be a compact operator on a Hilbert space H such that µn(T ) 6 C/n, n > 1, for some C > 0. Let λ1, λ2,... be the list of eigenvalues of T counted with multiplicities and such that |λ1| > |λ2| > ··· . Then

N 1 X 1 X τω(T ) = ω-lim λµT (λ) = ω-lim λi, (25) t→∞ log(1 + t) N→∞ log(1 + N) 1 i=1 λ∈σ(T ), λ/∈ t G where µT (λ) is the algebraic multiplicity of the eigenvalue λ and G is an arbitrary bounded neighbourhood of 0 ∈ C. Dixmier traces and some applications 1095

Here we emphasize that G is an arbitrary bounded neighbourhood of 0. In the case N = L (H ) the formula (25) says that we need to compute the sum of all 1 the eigenvalues λj(T ) not in the ‘squeezed’ neighbourhood t G as t → ∞, and then to apply the ω-limit. It is interesting to note the special case of (25) for measurable operators belonging to the ideal N(ψ1). The result below should be compared with the results of Theorems 6.6 and 6.7. Corollary 18.2. If T is a (Connes–Dixmier-) measurable operator satisfying the assumption of Theorem 18.1 (i) (respectively, (ii)), then

N 1 Z 1 X τω(T ) = lim λ dµT (λ) respectively, lim λi t→∞ log(1 + t) 1 N→∞ log(1 + N) λ/∈ t G i=1

∗ for any ω ∈ CD(R+). We ﬁx an orthonormal basis in H and identify each element x ∈ L (H ) with ∞ its matrix (xij)i,j=1. It is well known [49] that the triangular truncation operator T given by ( x , i j, T(x) = ij > 0, i < j, acts boundedly from the trace class L 1(H ) into L (1,∞)(H ). Noting that T(x) − diag(x) is quasi-nilpotent for every x ∈ L 1(H ), we obtain the following result. Corollary 18.3. The operator T(x) is Connes–Dixmier-measurable for every x ∈ 1 ∗ L (H ), and moreover τω(T(x)) = 0 for ω ∈ CD(R+). We conclude with the following result, which is an immediate consequence of Theorem 18.1 (see also Proposition 1 in [57]). Corollary 18.4. Let M be a compact n-dimensional Riemannian manifold and let T be a pseudodiﬀerential operator of order −n on M. Then

N 1 X τω(T ) = lim λk. N→∞ log N k=1

Bibliography

[1] J. Dixmier, “Existence de traces non normales”, C. R. Acad. Sci. Paris Sér. A 262 (1966), 1107–1108. [2] A. Connes, “The action functional in non-commutative geometry”, Comm. Math. Phys. 117:4 (1988), 673–683. [3] M. Wodzicki, “Noncommutative residue. I: Fundamentals”, K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., vol. 1289, Springer-Verlag, Berlin 1987, pp. 320–399. [4] M. Adler, “On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg–de Vries type equations”, Invent. Math. 50:3 (1978), 219–248. [5] Yu. I. Manin, “Algebraic aspects of nonlinear differential equations”, Current problems in mathematics, vol. 11, VINITI, Moscow 1978, pp. 5–152 (Russian); English transl., J. Soviet Math. 11 (1979), 1–122. 1096 A. L. Carey and F. A. Sukochev

[6] V. Guillemin, “A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues”, Adv. Math. 55:2 (1985), 131–160. [7] A. Connes, Noncommutative geometry, Academic Press, San Diego 1994. [8] J. M. Gracia-Bond´ıa,J. C. Várilly, and H. Figueroa, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Boston 2001. [9] A. Connes, “Essay on physics and noncommutative geometry”, The interface of mathematics and particle physics (Oxford, 1988), Inst. Math. Appl. Conf. Ser. New Ser., vol. 24, Oxford Univ. Press, New York 1990, pp. 9–48. [10] A. Connes and J. Lott, “Particle models and noncommutative geometry”, Recent advances in field theory (Annecy-le-Vieux, 1990), Nuclear Phys. B Proc. Suppl. 18 (1991), 29–47. [11] A. Connes, “Noncommutative geometry and reality”, J. Math. Phys. 36:11 (1995), 6194–6231. [12] W. Kalau, “Hamilton formalism in non-commutative geometry”, J. Geom. Phys. 18:4 (1996), 349–380. [13] D. Kastler, “Noncommutative geometry and basic physics”, Geometry and quantum physics (Schladming, 1999), Lecture Notes in Phys., vol. 543, Springer, Berlin 2000, pp. 131–230. [14] D. Kastler, “Noncommutative geometry and fundamental physical interaction: The Lagrangian level—historical sketch and description of the present situation”, J. Math. Phys. 41:6 (2000), 3867–3891. [15] A. Connes, “Trace de Dixmier, modules de Fredholm et géométrie riemannienne”, Conformal field theories and related topics (Annecy-le-Vieux, 1988), Nuclear Phys. B Proc. Suppl. 5:2 (1988), 65–70. [16] A. Connes, “Gravity coupled with matter and the foundation of non-commutative geometry”, Comm. Math. Phys. 182:1 (1996), 155–176. [17] M. F. Atiyah, “Elliptic operators, discrete groups and von Neumann algebras”, Colloque “Analyse et Topologie” en l’honneur de Henri Cartan (Orsay, 1974), Astérisque, vol. 32–33, Soc. Math. France, Paris 1976, pp. 43–72. [18] S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, “Singular traces and compact operators”, J. Funct. Anal. 137:2 (1996), 281–302. [19] S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, “Singular traces and nonstandard analysis”, Advances in analysis, probability and mathematical physics (Blaubeuren, 1992), Math. Appl., vol. 314, Kluwer, Dordrecht 1995, pp. 3–19. [20] S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, “Nonstandard representation of nonnormal traces”, Dynamics of complex and irregular systems (Bielefeld, 1991), Bielefeld Encount. Math. Phys., vol. VIII, World Scientific, River Edge, NJ 1993, pp. 1–11. [21] N. A. Azamov and F. A. Sukochev, “A Lidskii type formula for Dixmier traces”, C. R. Math. Acad. Sci. Paris 340:2 (2005), 107–112. [22] M.-T. Benameur and T. Fack, “Type II non-commutative geometry. I: Dixmier trace in von Neumann algebras”, Adv. Math. 199:1 (2006), 29–87. [23] A. L. Carey and J. Phillips, “Unbounded Fredholm modules and spectral flow”, Canad. J. Math. 50:4 (1998), 673–718. [24] A. L. Carey and J. Phillips, “Spectral flow in Fredholm modules, eta invariants and the JLO cocycle”, K-Theory 31:2 (2004), 135–194. [25] A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The Hochschild class of the Chern character for semifinite spectral triples”, J. Funct. Anal. 213:1 (2004), 111–153. [26] A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The local index formula in semifinite von Neumann algebras. I: Spectral flow”, Adv. Math. 202:2 (2006), 451–516. [27] A. L. Carey, J. Phillips, A. Rennie, and F. Sukochev, “The local index formula in semifinite von Neumann algebras. II: The even case”, Adv. Math. 202:2 (2006), 517–554. [28] A. L. Carey, J. Phillips, and F. A. Sukochev, “On unbounded p-summable Fredholm modules”, Adv. Math. 151:2 (2000), 140–163. [29] A. L. Carey, J. Phillips, and F. A. Sukochev, “Spectral flow and Dixmier traces”, Adv. Math. 173:1 (2003), 68–113. Dixmier traces and some applications 1097

[30] A. Connes, “Geometry from the spectral point of view”, Lett. Math. Phys. 34:3 (1995), 203–238. [31] P. G. Dodds, B. de Pagter, E. M. Semenov, and F. A. Sukochev, “Symmetric functionals and singular traces”, Positivity 2:1 (1998), 47–75. [32] K. Dykema, T. Figiel, G. Weiss, and M. Wodzicki, “Commutator structure of operator ideals”, Adv. Math. 185:1 (2004), 1–79. [33] D. Guido and T. Isola, “Singular traces for semifinite von Neumann algebras”, J. Funct. Anal. 134:2 (1995), 451–485. [34] D. Guido and T. Isola, “On the domain of singular traces”, Internat. J. Math. 13:6 (2002), 667–674. [35] N. Kalton and F. Sukochev, “Rearrangement invariant functionals with applications to traces on symmetrically normed ideals”, Canad. Math. Bull. (to appear). [36] R. Prinzis, Traces residuelles et asymptotique du spectre d’operateurs pseudo-differentiels, Thèse,Universitéde Lyon 1994. [37] D. Guido and T. Isola, “Dimensions and singular traces for spectral triples, with applications to fractals”, J. Funct. Anal. 203:2 (2003), 362–400. [38] D. Guido and T. Isola, “Singular traces and their applications to geometry”, Operator algebras and quantum field theory (Rome, 1996), Internat. Press, Cambridge, MA 1997, pp. 440–456. [39] F. A. Sukochev, “Operator estimates for Fredholm modules”, Canad. J. Math. 52:4 (2000), 849–896. [40] P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, and F. A. Sukochev, “Singular symmetric functionals”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI ) 290 (2002), 42–71 (Russian); English transl., J. Math. Sci. (New York) 124:2 (2004), 4867–4885. [41] P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, and F. A. Sukochev, “Singular symmetric functionals and Banach limits with additional invariance properties”, Izv. Ross. Akad. Nauk Ser. Mat. 67:6 (2003), 111–136 (Russian); English transl., Izv. Math. 67:6 (2003), 1187–1212. [42] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of linear operators, Nauka, Moscow 1978 (Russian); English transl., Transl. Math. Monogr., vol. 54, Amer. Math. Soc., Providence, RI 1982. [43] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I: Sequence spaces, Ergeb. Math. Grenzgeb., vol. 92, Springer-Verlag, Berlin 1977. [44] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II: Function spaces, Ergeb. Math. Grenzgeb., vol. 97, Springer-Verlag, Berlin 1979. [45] M. Sh. Braverman and A. A. Mekler, “The Hardy–Littlewood property for symmetric spaces”, Sibirsk. Mat. Zˇ. 18:3 (1977), 522–540 (Russian); English transl., Siberian Math. J. 18:3 (1977), 371–385. [46] W. Rudin, Principles of mathematical analysis, McGraw-Hill, New York 1964. [47] T. Fack and H. Kosaki, “Generalized s-numbers of τ-measurable operators”, Pacific J. Math. 123:2 (1986), 269–300. [48] T. Fack, “Sur la notion de valeur caractéristique”, J. Operator Theory 7:2 (1982), 307–333. [49] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Nauka, Moscow 1965 (Russian); English transl., Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI 1969. [50] F. A. Sukochev and V. I. Chilin, “Symmetric spaces on semifinite von Neumann algebras”, Dokl. Akad. Nauk SSSR 313:4 (1990), 811–815 (Russian); English transl., Soviet Math. Dokl. 42:1 (1991), 97–101. [51] V. I. Chilin and F. A. Sukochev, “Weak convergence in non-commutative symmetric spaces”, J. Operator Theory 31:1 (1994), 35–65. [52] P. G. Dodds, T. K. Dodds, and B. de Pagter, “Fully symmetric operator spaces”, Integral Equations Operator Theory 15:6 (1992), 942–972. [53] P. G. Dodds, T. K.-Y. Dodds, and B. de Pagter, “Non-commutative Banach function spaces”, Math. Z. 201:4 (1989), 583–597. 1098 A. L. Carey and F. A. Sukochev

[54] N. Kalton, “Spectral characterization of sums of commutators, I”, J. Reine Angew. Math. 504 (1998), 115–125. [55] K. J. Dykema and N. J. Kalton, “Spectral characterization of sums of commutators, II”, J. Reine Angew. Math. 504 (1998), 127–137. [56] K. J. Dykema and N. J. Kalton, “Sums of commutators in ideals and modules of type II factors”, Ann. Inst. Fourier (Grenoble) 55:3 (2005), 931–971. [57] T. Fack, “Sums of commutators in non-commutative Banach function spaces”, J. Funct. Anal. 207:2 (2004), 358–398. [58] G. G. Lorentz, “A contribution to the theory of divergent sequences”, Acta Math. 80:1 (1960), 167–190. [59] M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, Academic Press, New York–London 1972. [60] S. Lord, A. Sedaev, and F. Sukochev, “Dixmier traces as singular symmetric functionals and applications to measurable operators”, J. Funct. Anal. 224:1 (2005), 72–106. [61] F. Cipriani, D. Guido, and S. Scarlatti, “A remark on trace properties of K-cycles”, J. Operator Theory 35:1 (1996), 179–189. [62] J. V. Varga, “Traces on irregular ideals”, Proc. Amer. Math. Soc. 107:3 (1989), 715–723. [63] G. H. Hardy, Divergent series, Clarendon Press, Oxford 1949. [64] L. Sucheston, “Banach limits”, Amer. Math. Monthly 74 (1967), 308–311. [65] A. Connes, “Entire cyclic cohomology of Banach algebras and characters of θ-summable Fredholm modules”, K-Theory 1:6 (1988), 519–548. [66] A. Connes, “Compact metric spaces, Fredholm modules, and hyperfiniteness”, Ergodic Theory Dynam. Systems 9:2 (1989), 207–220. [67] A. Connes, “Non-commutative differential geometry”, Inst. Hautes Etudes´ Sci. Publ. Math. 62 (1985), 41–144. [68] M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian geometry, I”, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. [69] M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian geometry, III”, Math. Proc. Cambridge Philos. Soc. 79:1 (1976), 71–99. [70] E. Schrohe, M. Walze, and J.-M. Warzecha, “Construction de triplets spectraux àpartir de modules de Fredholm”, C. R. Acad. Sci. Paris Sér. I Math. 326:10 (1998), 1195–1199. [71] A. Rennie, “Smoothness and locality for nonunital spectral triples”, K-Theory 28:2 (2003), 127–165. [72] J. Phillips, “Self-adjoint Fredholm operators and spectral flow”, Canad. Math. Bull. 39:4 (1996), 460–467. [73] J. Phillips, “Spectral flow in type I and type II factors—a new approach”, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun., vol. 17, Amer. Math. Soc., Providence, RI 1997, pp. 137–153. [74] J. Phillips and I. Raeburn, “An index theorem for Toeplitz operators with noncommutative symbol space”, J. Funct. Anal. 120:2 (1994), 239–263. [75] M. Breuer, “Fredholm theories in von Neumann algebras, I”, Math. Ann. 178 (1968), 243–254. [76] M. Breuer, “Fredholm theories in von Neumann algebras, II”, Math. Ann. 180:4 (1969), 313–325. [77] V. S. Perera, Real-valued spectral flow in a type II∞ factor, Ph.D. Thesis, IUPUI, Indianapolis 1993. [78] V. S. Perera, “Real-valued spectral flow”, Multivariable operator theory (Seattle, WA, 1993), Contemp. Math., vol. 185, Amer. Math. Soc., Providence, RI 1995, pp. 307–318. [79] M. T. Benameur, A. L. Carey, J. Phillips, A. Rennie, F. A. Sukochev, and K. P. Wojciechowski, “An analytic approach to spectral flow in von Neumann algebras”, Analysis, geometry and topology of elliptic operators (Roskilde, 2005) (B. Booß-Bavnbek et al., eds.), Papers in honor of Krzysztof P. Wojciechowski, Proceedings of a workshop, World Scientific, Singapore 2006, pp. 297–352. [80] B. Booß-Bavnbek, M. Lesch, and J. Phillips, “Unbounded Fredholm operators and spectral Flow”, Canad. J. Math. 57:2 (2005), 225–250. Dixmier traces and some applications 1099

[81] V. Mathai, “Spectral ﬂow, eta invariants, and von Neumann algebras”, J. Funct. Anal. 109:2 (1992), 442–456.

[82] V. S. Perera, “Real valued spectral flow in a type II∞ factor”, Houston J. Math. 25:1 (1999), 55–66. [83] E. Getzler, “The odd Chern character in cyclic homology and spectral flow”, Topology 32:3 (1993), 489–507. [84] J.-L. Loday, Cyclic homology, Grundlehren Math. Wiss., vol. 301, Springer-Verlag, Berlin 1998. [85] A. Connes and H. Moscovici, “The local index formula in noncommutative geometry”, Geom. Funct. Anal. 5:2 (1995), 174–243. [86] O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics. vol. 1. C∗- and W ∗-algebras, algebras, symmetry groups, decomposition of states, Texts and Monographs in Physics, Springer-Verlag, New York–Heidelberg 1979. [87] L. A. Coburn, R. G. Douglas, D. G. Schaeffer, and I. M. Singer, “C∗-algebras of operators on a half-space. II: Index theory”, Inst. Hautes Etudes´ Sci. Publ. Math. 40 (1971), 69–79. [88] M. Lesch, “On the index of the infinitesimal generator of a flow”, J. Operator Theory 26:1 (1991), 73–92. [89] N. Higson, “The local index formula in noncommutative geometry”, Contemporary developments in algebraic K-theory, ICTP Lect. Notes, vol. XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste 2004, pp. 443–536. [90] A. Connes, “Sur la théorie non commutative de l’intégration”, Algèbres d’opérateurs (Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, Springer-Verlag, Berlin 1979, pp. 19–143. [91] M. A. Shubin, “Pseudodifferential almost-periodic operators and von Neumann algebras”, Trudy Moskov. Mat. Obˇsˇc. 35 (1976), 103–164 (Russian); English transl., Trans. Moscow Math. Soc., 1979, no. 1, 103–166. [92] M. A. Shubin, “The spectral theory and the index of elliptic operators with almost periodic coefficients”, Uspekhi Mat. Nauk 34:2 (1979), 95–135 (Russian); English transl., Russian Math. Surveys 34:2 (1979), 109–157. [93] J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Cahier Scientifiques, vol. XXV, Gauthier–Villars, Paris 1969. [94] M. Farber and S. Weinberger, “On the zero-in-the-spectrum conjecture”, Ann. of Math. (2) 154:1 (2001), 139–154. [95] L. G. Brown, “Lidskii’s theorem in the type II case”, Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow 1986, pp. 1–35.

A. L. Carey Received 10/NOV/05 Mathematical Sciences Institute, Translated by A. L. CAREY Australian National University and F. A. SUKOCHEV E-mail: [email protected] F. A. Sukochev School of Informatics and Engineering, Flinders University E-mail: [email protected]