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Dixmier Traces and Some Applications in Non-Commutative Geometry

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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 pseudodiffer- ential 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 expres- sion 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 integra- tion 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 Ces`aromean 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 mea- surable 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.
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