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Subclasses of Dixmier (Singular) Traces and Related Classes of Measurable Operators

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Subclasses of Dixmier (Singular) Traces and Related Classes of Measurable Operators Subclasses of Dixmier (singular) traces and related classes of measurable operators by Alexandr Usachev A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales. School of Mathematics and Statistics Faculty of Science August 2013 Abstract In the framework of noncommutative geometry the singular (Dixmier) traces, originally introduced by J. Dixmier in 1966, have become an indispensable tool. These traces are defined via dilation invariant extended limits on the space of bounded measurable functions. Important results in noncommutative geometry (e.g. Connes Character Theorem, relation between Dixmier traces, heat function- als and zeta-function residues) are proved under various additional conditions on these extended limits. Every such condition distinguishes a subclass of Dixmier traces. The present thesis studies the relation between these classes and an important concept of measurable operators (with respect to these subclasses) introduced in noncom- mutative geometry by A. Connes in 1988. In most cases we provide new char- acterisations of measurability and definitive description of classes of measurable operators. i Acknowledgements I would like to thank my supervisor Prof. Fedor Sukochev, for all the effort which he has put into me. I would like to thank him for introducing me to interesting problems and letting me work independently while also providing help whenever I needed. I would also like to thank my colleagues and co-authors E. M. Semenov, A. A. Sedaev, D. Potapov, D. Zanin for sharing their thoughts and working together. ii Contents 1 Overview 1 1.1 Introduction . 1 1.2 Structure of the thesis . 9 2 Extended limits 13 2.1 Extended limits and limits along an ultrafilter on l1 . 13 2.2 Translation invariant extended limits on l1 (Banach limits) . 17 2.3 Extended limits on L1 ........................ 25 2.4 Translation invariant extended limits on L1 . 28 3 Dilation invariant extended limits and Dixmier trace construc- tion 38 3.1 Dilation invariant extended limits . 38 3.2 Dixmier traces . 41 3.2.1 Lorentz ideals . 42 3.2.2 Construction of Dixmier traces . 44 4 Connes-Dixmier traces 60 4.1 The classes of Dixmier and Connes-Dixmier traces are distinct . 61 4.2 The classes of Dixmier and Connes-Dixmier measurable operators are distinct . 64 5 Extended limits with additional invariance properties and Dixmier traces 69 5.1 Ces`aroinvariant extended limits . 70 iii 5.2 On measurability with respect to the subclass DM of Dixmier traces 75 5.3 Dilation and Ces`aro-invariant extended limits . 80 6 Exponentiation invariant extended limits 86 6.1 Dixmier traces generated by exponentiation invariant extended limits 86 6.2 Lidskii formula for Dixmier traces generated by exponentiation invariant extended limits . 93 6.3 Zeta-function residues and Dixmier traces . 99 References 111 iv Chapter 1 Overview 1.1 Introduction Alain Connes' noncommutative geometry is a relatively young discipline foun- ded some thirty years ago, but it is rapidly developing both theory and applica- tions (see e.g. [21, 25, 26, 42]). In all of them Dixmier traces play an important role as a direct analogue of integral in the noncommutative context. Consider the algebra B(H) of all bounded linear operators on a separable Hilbert space H and the ideal M1;1 of all compact operators in B(H), whose partial sums of singular values diverge logarithmically, that is ( ) − 1 Xn 1 M1;1 := A 2 B(H) is compact : sup µ(k; A) < 1 ; n≥1 log(1 + n) k=0 where fµ(n; A)gn≥0 is the sequence of singular values of a compact operator A 2 B(H). Initially, for an arbitrary translation and dilation invariant extended limit ! on l1, J. Dixmier [28] constructed the weight 0( ) 1 Xn−1 @ 1 A Tr (A) := ! µ(k; A) ; 0 ≤ A 2 M 1: (1.1) ! log(1 + n) 1; k=0 n≥1 The weight Tr! extends by linearity to a trace on M1;1, which fails to be normal (in fact, J. Dixmier worked in more general setting of the Lorentz ideal M of compact operators which partial sums of singular values diverge as a given concave function : (0; 1) ! (0; 1)). That was the first example of singular (that is, non-normal) trace and for some twenty years Dixmier traces were nothing but 1 a \pathological monsters"1. Only in 1985 A. Connes \obtained from a singular traces applications which were not monstrous at all"1. Those applications were in the area of noncommutative geometry [20]. The basic object of noncommutative geometry is a spectral triple (A; H; D) (see e.g. [21, Chapter IV]), where A is a unital non-degenerate ∗-algebra of bounded linear operators on a separable complex Hilbert space H, D : DomD ! H is a selfadjoint linear operator with the properties Da − aD 2 B(H) for all a 2 A and hDi−1 := (1 + D2)−1=2 is compact. Compact operators play the role of infinitesimals in Connes' quantised calcu- lus [21, Chapter IV]. In particular, the compact operator hDi−1 is the analogue of an infinitesimal length element ds (see e.g. [21, p. 545]) and the integral of a 2 A is given as follows −n Tr!(ahDi ); (1.2) where n (assumed to exist) is the smallest value such that hDi−p is trace class for p > n, and Tr! is a Dixmier trace. Applications of Dixmier traces as a substitute of an integral in non-classical spaces range from fractals [45,56], to foliations [7], to spaces of noncommuting co- ordinates [22,39] and applications in string theory and Yang-Mills [20,24,32,81], Einstein-Hilbert actions, particle physics' standard model [18,23,53] and quantum field theory [18,20,22]. Although in most applications the logarithmic Dixmier traces (and the ideal M1;1 as their domain) appear, the arbitrary Lorentz ideals ( ) − 1 Xn 1 M := A 2 B(H) is compact : sup µ(k;A) < 1 n≥1 (n) k=0 (where : (0; 1) ! (0; 1) is a given concave function) also play an important role and appear in applications. For instance, in [66] Dixmier traces, which sum 2 2 \log -divergence" (and the ideal M with (t) = log (1 + t), t > 0 as their domain) were used to study pseudodifferential operators of a special type. The latter ideal also appeared in [33], when authors described a spectral asymptotic of 1J. Dixmier's letter to the conference \Singular traces and their applications" at Luminy, 2012 (see e.g. [60]) 2 a twisted bi-Laplacian. Note that, such an aspect of noncommutative geometry as the notion of the spectral triple, is broad enough to treat also commutative singular spaces (such as fractals), which are too irregular to be treated with the in- struments of Riemannian geometry. In [44{46] authors employed general Dixmier traces (that is, Dixmier traces on general Lorentz ideals M ) to obtain a reason- able description of the dimension of fractals. The general Lorentz ideals M also appeared in a recent paper by V. Gayral and F. Sukochev [40], generalising the results of [66] on pseudodifferential operators. Despite the fact that there are infinitely many Dixmier traces, there are op- erators (classical pseudodifferential operators, for instance) on which all Dixmier traces coincide. Such operators are called Dixmier measurable. The origin of this term becomes clear from the formula (1.2). Indeed, the formula (1.2) (with a fixed operator a 2 A), in general, gives different values for different Dixmier traces. However, for measurable operators the formula (1.2) yields a unique value, which is the value of an integral at a 2 A. In early 1980's M. Wodzicki [101] discovered the only trace on the algebra of classical pseudodifferential operators. He proved that the noncommutative (Wodzicki) residue on this algebra has tracial properties. In 1988 A. Connes showed the connection between the Dixmier trace(s) and the M. Wodzicki's non- commutative residue in differential geometry [20]. This result, known as Connes' Trace Theorem, has become the cornerstone of noncommutative integration `ala Connes. Connes' Trace Theorem associates a spectral and essentially incalculable object to a calculable formula. If A is a classical pseudodifferential operator of order −d on a d-dimensional closed manifold, then the extension of A, as an oper- ator from square integrable functions to square integrable functions, is a compact operator that belongs to the domain of a Dixmier trace (that is, to the ideal M1;1) and Tr!(A) = ResW (A): The natural extension of this result for non-classical pseudodifferential oper- ators of order −d and the vector-valued extension of Wodzicki residue is given in [50]. 3 Also in [20] A. Connes described in part the relationship between Dixmier traces, heat kernel asymptotics and ζ-function at their leading singularity. This relationship plays an important role in noncommutative geometry. Indeed, in a recent studies of noncommutative spaces [16,17,67,68] determination of Dixmier summability of an operator (that is, showing that an operator belongs to the cor- responding Lorentz ideal M ) relies usually either on Weyl's theorem or Cwickel type inequalities. In particular, for operators arising from \noncommutative ac- tion principle" (that is, when one minimises functionals on noncommutative al- gebras), the (classical) geometric context is not guaranteed to exist. So Weyl's theorem is of no use. The relation of Dixmier traces to ζ-function residues and heat kernel type expansions resolve this obstacle. These alternative techniques are likely to be relevant for (very) noncommutative and physically inspired examples. In partic- ular, in [11] the ζ-function approach to Dixmier traces is used to derive a general formula for the Chern character of an M1;1 summable Breuer-Fredholm module.
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