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

Subclasses of Dixmier (singular) traces and related classes of measurable operators

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 functionals 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 noncommutative 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 l∞ ...... 13

2.2 Translation invariant extended limits on l∞ (Banach limits) . . . . 17

2.3 Extended limits on L∞ ...... 25

2.4 Translation invariant extended limits on L∞ ...... 28

3 Dilation invariant extended limits and Dixmier trace construction 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 applications (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,∞ of all compact operators in B(H), whose partial sums of singular values diverge logarithmically, that is ( ) − 1 Xn 1 M1,∞ := A ∈ B(H) is compact : sup µ(k, A) < ∞ , n≥1 log(1 + n) k=0 where {µ(n, A)}n≥0 is the sequence of singular values of a compact operator A ∈ B(H). Initially, for an arbitrary translation and dilation invariant extended limit

ω on l∞, J. Dixmier [28] constructed the weight ( )  Xn−1  1  Tr (A) := ω µ(k, A) , 0 ≤ A ∈ M ∞. (1.1) ω log(1 + n) 1, k=0 n≥1

The weight Trω extends by linearity to a trace on M1,∞, which fails to be normal

(in fact, J. Dixmier worked in more general setting of the Lorentz ideal Mψ of compact operators which partial sums of singular values diverge as a given concave function ψ : (0, ∞) → (0, ∞)). 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 ∈ B(H) for all a ∈ A and hDi−1 := (1 + D2)−1/2 is compact. Compact operators play the role of inﬁnitesimals in Connes’ quantised calculus [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 ∈ A is given as follows

−n Trω(ahDi ), (1.2) where n (assumed to exist) is the smallest value such that hDi−p is trace class for p > n, and Trω 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,∞ as their domain) appear, the arbitrary Lorentz ideals ( ) − 1 Xn 1 Mψ := A ∈ B(H) is compact : sup µ(k,A) < ∞ n≥1 ψ(n) k=0 (where ψ : (0, ∞) → (0, ∞) 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 Mψ with ψ(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 Mψ) to obtain a reason- able description of the dimension of fractals. The general Lorentz ideals Mψ 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 operators (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 ∈ 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 ∈ 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 noncommutative residue in differential geometry [20]. This result, known as Connes’ Trace Theorem, has become the cornerstone of noncommutative integration àla 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 operator 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,∞) and

Trω(A) = ResW (A).

The natural extension of this result for non-classical pseudodiﬀerential operators 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 corresponding Lorentz ideal Mψ) relies usually either on Weyl’s theorem or Cwickel type inequalities. In particular, for operators arising from “noncommutative action principle” (that is, when one minimises functionals on noncommutative algebras), 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 particular, in [11] the ζ-function approach to Dixmier traces is used to derive a general formula for the Chern character of an M1,∞ summable Breuer-Fredholm module. J. Dixmier’s construction of a non-normal trace was not the only one. Un- knowingly of the paper by J. Dixmier [28], A. Pietsch approached the problem of existence of singular traces on operator ideals (in the framework of his general theory of operator ideals) from the axiomatic point of view. The main open problem suggested by Pietsch was “Whether there exist quasi-Banach operator ideals that support different continuous traces?” A solution to this problem was given by Nigel Kalton and can be found in his paper [48]. Pietsch’s recent paper [72] details a complete construction of traces on the ideal M1,∞ and its Banach space analogue. In the paper [99], J. Varga characterized those compact operators A such that certain ideals associated with A admit nontrivial traces. In fact, this P M n construction it equivalent to that of Lorentz ideal ψ with ψ(n) = k=0 µ(k, A), n ≥ 0. For an exposition of Varga’s approach and its extension to general semifi- nite von Neumann algebras, we refer to [30, 31] and [43]. In the paper [102], M. Wodzicki suggested to transfer Dixmier’s construction of a singular trace to an arbitrary operator ideal. Questions of finiteness and linearity become very hard in this general setting. The paper [93] generalises some of the results from [102].

4 These results among many others form the independent area of research, that is, the theory of singular traces to which this thesis also belongs.

A part of this thesis studies an important concept of measurable operators introduced in noncommutative geometry by A. Connes in [20] and [21, Definition 7, IV.2.β]. We begin by describing most frequently used classes of Dixmier traces which play a role in the definition of the latter concept. The class of all Dixmier traces is denoted by D. The construction of Dixmier traces as given in (1.1) involves the extended limit ω (that is, a singular state on the algebra l∞ of all bounded sequences). Depending on the additional properties of this state, there are several natural subclasses of Dixmier traces in noncommutative geometry. First, it was observed by A. Connes [21], that in order to ensure that the functional Trω given by (1.1) extends to a positive linear functional on M1,∞, it is sufficient to only assume that ω is a dilation invariant extended limit on l∞. A natural way to generate such ω’s was suggested by A. Connes [21, Section

IV, 2β] by observing that for any extended limit γ on l∞ a functional ω := γ ◦ M is dilation invariant. Here, the bounded operator M : l∞ → l∞ is given by the formula Xn 1 x − (Mx) = k 1 , n ≥ 1. n log(1 + n) k k=1

The class C of all Dixmier traces Trω deﬁned by such ω’s was termed Connes- Dixmier in [58]. Finally, various important formulas of noncommutative geometry, like the Connes’ formula for a representative of the Hochschild class of the Chern character for (p, ∞)-summable spectral triples (see e.g. [10, Theorem 7] and [7, Theorem 6]) as well as those involving heat kernel estimates and generalised ζ-function residues

(see e.g. [7, 10–12, 94]) are frequently established for yet a smaller subset DM of Connes-Dixmier traces, when the extended limit ω is assumed to be M-invariant, that is when ω = ω ◦ M.

This class of extended limits was ﬁrst introduced in [11] (see also [30]) and further

5 studied and used in [4,7,12]. Also in [10, 11] the results on heat kernel asymptotics and generalised ζ- function residues were established for Dixmier traces Trω, provided that ω was additionally chosen to be exponentiation invariant (see relevant deﬁnitions in subsequent sections). Indeed, in [11] formulas

Trω = ζω◦log = ξω, (1.3)

were proved for M- and exponentiation invariant extended limits ω. Here ζγ is a zeta-function residue and ξω is a heat kernel functional. In [10] these assumptions were relaxed to dilation and exponentiation invariance of ω. In view of this variety of subclasses of Dixmier traces, it is natural to ask the question:

How do the subclasses of Dixmier traces relate to one another?

Since in most geometric applications the operators are “measurable”, the following specialisation of the question above is of a particular interest:

How do the sets of operators measurable with respect to diﬀerent subclasses of Dixmier traces relate to one another?

We now introduce a rigorous deﬁnition of measurability, together with the notation for the classes of measurable operators. We formulate it for a general

Lorentz ideal Mψ of compact operators whose partial sums of singular values diverge as a given concave function ψ (see a rigorous deﬁnition of the ideal Mψ in subsequent sections).

Deﬁnition 1.1.1. Let A denote a subclass of Dixmier traces on Mψ. The set MA A ∈ M ψ of all -measurable operators consists of all elements A ψ such that

Trω(A) takes the same value for all Trω ∈ A.

We refer to [13, 21, 58, 59, 77, 92] for a discussion of properties and concrete examples of C and D-measurable elements, although we point out that the precise relationship between these classes was not clear till recent time. A priori, we have

DM ⊆ C ⊆ D (1.4)

6 and hence, trivially, D MD ⊆ MC ⊆ M M ψ ψ ψ . (1.5)

In Sections 4 and 5 we show that all inclusions in (1.4) and (1.5) are proper. One of the main results of Chapter 3 gives a complete characterisation of the set of all D-measurable elements from Mψ.

Theorem 1.1.2. An operator A ∈ Mψ is measurable with respect to the set of all Dixmier traces if and only if the limit

bαkc 1 Xn 1 X lim λ(i, A) n→∞ log(1 + n) kψ(αk) k=1 i=0 exists uniformly in α ≥ 1, where {λ(n, A)}n≥0 is the sequence of eigenvalues of the operator A.

This theorem signiﬁcantly extends [58, Theorem 6.6] (in a rather unexpected way) asserting that an arbitrary positive operator A from Mψ is D-measurable if and only if it is Tauberian, that is there exists the limit

− 1 Xn 1 lim λ(k, A). n→∞ log(1 + n) k=0 In Corollary 3.2.17 below, we present a short new proof of this result as a consequence of Theorem 1.1.2. In Chapter 4 we investigate the relationship between the classes of Dixmier and Connes-Dixmier traces. We show that they diﬀer even on the classical

Dixmier ideal M1,∞. Recently the distinction between C and D was studied by A. Pietsch in terms of density characters (see [69,70,72]), where the advanced techniques were developed. Without any doubts, these techniques are of a wider interest in the theory of singular traces. Our approach is completely diﬀerent from that of A. Pietsch and the proof provided here is much shorter. Notwithstanding the diﬀerence between the sets of Dixmier and Connes-

Dixmier traces, it is known that a positive operator A ∈ M1,∞ is Connes-Dixmier measurable if and only if it is Dixmier measurable [58, Corollary 3.9]. This result naturally raises the question, whether for an arbitrary concave function ψ the Connes-Dixmier measurability is equivalent to Dixmier measurability on the

7 cone of all positive elements from Mψ. The second main result of Chapter 4 (Theorem 4.2.3 below) shows that the answer is (surprisingly) negative. That is, we prove that there is a Lorentz ideal Mψ such that these two classes of traces generate distinct sets of measurable elements.

In Chapter 5 we study the class DM of Dixmier traces generated by M- invariant extended limits. The main result of this chapter shows that the set of all

Dixmier measurable elements is a proper subset of the set of all DM -measurable elements. As an immediate corollary, we conclude that the set DM is a proper subset of the set of all Dixmier traces. As it was pointed out, various important formulae of noncommutative geometry were established for dilation and exponentiation invariant extended limits ω. The former assumption was needed in order that the formula (1.1) deﬁnes a Dixmier trace. In Chapter 6 we prove that one can apply the Dixmier construction (1.1) for every exponentiation invariant extended limit ω instead of dilation invariant extended limit and still obtain a non-normal trace (see Theorem 6.1.4 below). We also show that these singular traces are, in fact, Dixmier traces. That is, for every exponentiation invariant extended limit ω there exists a dilation invariant extended limit ω1 such that Trω = Trω1 . We denote this “new” subclass of

Dixmier traces by DP .

We prove (see Theorem 6.1.8 below) that the class of DP -measurable operators is strictly wider than the class of Dixmier measurable operators. In particular,

DP is a proper subset of D. We stress out that the computation of a Dixmier trace of an operator A can be a highly non-trivial task due to the fact, that in general one has no access to the singular values of A. A remedy to this problem is provided by the Lidskii formula, which allows to compute the trace using the eigenvalues of the operator, instead of its singular values. The classical Lidskii Theorem asserts that X∞ Tr(A) = λ(n, A) n=0 for any trace class operator A. Here, {λ(n, A)}n≥0 is the sequence of eigenvalues of

8 A (counting with algebraic multiplicities), taken in an arbitrary order. This arbi- P ∞ | | trariness of the order is due to absolute convergence of the series n=0 λ(n, A) . In particular, we can choose a decreasing order of |λ(n, A)|. The core diﬀerence of this situation with the setting of Dixmier traces living on P M ∞ | | ∈ M the ideal 1,∞ is that in general the series n=0 λ(n, A) diverges for A 1,∞ ∈ M 1 ≥ (take for example A 1,∞ such that λ(n, A) = n+1 , n 0). The following theorem yields an analogue of Lidskii formula for Connes- Dixmier traces.

Theorem 1.1.3 ( [80, Theorem 2]). Let Trω be a Connes-Dixmier trace on M1,∞.

We have   X  1  Tr (A) = ω λ , A ∈ M ∞, (1.6) ω log(1 + n) 1, ∈ | | 1 λ σ(A): λ > n where σ(A) denotes the spectrum of the operator A.

It was also shown in [80, Theorem 5] that there exists a Dixmier trace Trω such that the formula (1.6) does not hold. For Dixmier traces the summation method should be modiﬁed as follows.

Theorem 1.1.4 ( [80, Theorem 3]). Let Trω be a Dixmier trace on M1,∞. We have   X  1  Tr (A) = ω λ , A ∈ M ∞, (1.7) ω log(1 + n) 1, ∈ | | log n λ σ(A): λ > n where σ(A) denotes the spectrum of an operator A.

One of the main results of Section 6 (see Theorem 6.2.7 below) asserts that the formula (1.6) holds for every trace Trω ∈ DP . We also investigate the relation of “new” traces to zeta-functions residues and prove that every trace Trω ∈ DP is in fact a zeta-function residue. This result generalises the formula (1.3) by showing that the assertion of dilation invariance of ω is redundant.

1.2 Structure of the thesis

As it was stated above the construction of Dixmier traces strongly depends on the choice of extended limits. Moreover, most of the problems concerning

9 measurability are reduced to that of extended limits. So, extended limits play an important role in the theory of singular traces. In Chapter 2 we introduce the concept of translation invariant extended limit on the space l∞ of bounded sequences (Banach limits). We state some classical and new results on Banach limits. In his original construction (1.1) J. Dixmier worked in discrete setting of the space l∞, that is, he considered extended limits on the sequence space l∞. However, it was proved to be useful to work in continuous setting, that is, to consider extended limits on the function space L∞, rather than that on the sequence space l∞ (see e.g. [10,11,21,90–92]). This fact has led to extension of the notion of Banach limit and introduction of the concept of translation invariant extended limit on the space L∞ of (classes of) essentially bounded real-valued functions (see also [58]). In Chapter 3 we describe the relation between translation and dilation invariant extended limits and construct Dixmier traces using dilation invariant extended limits on L∞. Although our setting is distinct from that of J. Dixmier, we show that our “continuous” construction yields the same set of traces as the original “discrete” construction (1.1). One of the main results of this section is a complete characterisation of Dixmier measurable operators from Mψ (for every ideal Mψ, which admits non-trivial Dixmier traces). We present this result in terms of operator singular values and (in a more computable form) in terms of operator eigenvalues. Chapter 4 studies the relationship between the class D of all Dixmier traces and the class C of Connes-Dixmier traces. The study shows that they diﬀer even on the classical Dixmier ideal M1,∞. Furthermore, it proves that there is a

Lorentz ideal Mψ, such that these two classes of traces generate distinct sets of measurable elements. In Chapter 5 we consider a smaller class of Dixmier traces, which are generated by extended limits with additional (besides dilation) invariance properties. In particular, we show that in addition to dilation invariance one can assume Ces`aro H-invariance (which also implies translation invariance) of extended limit and still obtain the whole class of Dixmier traces.

10 Also in this chapter we consider M-invariant extended limits and show that M-invariance implies H-, dilation and translation invariance of extended limit. However, the class of H-, dilation and translation invariant extended limits is wider than that of M-invariant extended limits. It is shown in Theorem 5.2.2 below that the class DM of Dixmier traces generated by M-invariant extended limits is a proper subclass of all Dixmier traces. In Chapter 6 we introduce new class of traces, applying Dixmier’s construction (1.1) to exponentiation invariant extended limit, instead of dilation invariant one. First of all we show that this construction produces traces on M1,∞, which are Dixmier traces. Next, we construct an example, which shows that these “new” traces constitute a proper subset DP in the set of all Dixmier traces. Finally, we prove the Lidskii formula for traces from DP and show that every trace from DP is, in fact, a zeta-function residue.

Most of the results presented in this thesis are published in the following papers: 1. E. Semenov, F. Sukochev, A. Usachev Structural properties of the Banach limits set, Doklady Mathematics, 84:3 (2011) 802–803.

2. F. Sukochev, A. Usachev, D. Zanin On the distinction between the classes of Dixmier and Connes-Dixmier traces, Proc. Amer. Math. Soc., 141:6 (2013), 2169–2179.

3. F. Sukochev, A. Usachev, D. Zanin Generalized limits with additional invariance properties and their applications to noncommutative geometry, Advances in Mathematics, 239 (2013) 164–189.

4. F. Sukochev, A. Usachev, D. Zanin Dixmier traces generated by exponentiation invariant generalised limits, accepted to Journal of Noncommutative Geometry on 01/05/12.

5. E. Semenov, F. Sukochev, A. Usachev Geometric properties of the set of

11 Banach limits, accepted to Izvestiya Mathematics on 01/06/13.

Some results of this thesis were presented at the following conferences: 1. A. Usachev, Fourier-Haar coeﬃcients of the function from Marcienkiewicz space, International conference “Banach Spaces Geometry”, 2010, St.Petersburg, Russia.

2. A. Usachev, Generalized limits with additional invariance properties and their applications, “Singular Traces and their Applications”, 2012, Marseilles, France.

3. A. Usachev, Invariant generalized limits and their applications to noncommutative geometry Voronezh Winter School, VSU, Voronezh, Russia - 2012.

4. A. Usachev, On the distinction between the classes of Dixmier and Connes- Dixmier traces ICOMAS-2012, University of Memphis, TN, USA.

5. A. Usachev, On the distinction between Dixmier and Connes-Dixmier measurability IWOTA-2012, University of New South Wales, Sydney, Australia.

12 Chapter 2

Extended limits

In the present chapter we outline the diﬀerent approaches to the extended limits on l∞ and L∞. We discuss the important properties of extended limits, which shall be used in subsequent chapters.

2.1 Extended limits and limits along an ultra-

ﬁlter on l∞

Recall that by l∞ we denote the Banach space of all bounded sequences x =

(x0, x1,...) with the norm k k | | x l∞ := sup xn . n≥0

Deﬁnition 2.1.1. A positive linear functional on l∞ which equals the ordinary limit on convergent sequences is said to be an extended limit.

The existence of extended limits follows from Hanh-Banach theorem (see e.g. [5, Chapter II, §1, Theorem 1]). Extended limits yield a natural extension of a limit functional deﬁned on the space of all convergent sequences to the space of all bounded sequences. Like the limit superior and limit inferior extended limits can be applied in situations where one wants to algebraically manipulate with limit equations or inequalities, even when it is not assured beforehand that the limit (in the classical sense) exists.

13 Extended limits provide an example of a continuous linear functionals on l∞ which can not be written in an “integral form” X∞ l(x) = xnyn, x ∈ l∞, (2.1) n=0 for some ﬁxed element y ∈ l1.

Via mapping (2.1) the space l1 can be identiﬁed with the band (that is, an order complete ideal) of all order continuous linear functionals on l∞ [3, Theorem

d ∗ 9.34]. Denoting by l1 the disjoint complement of this band we obtain l∞ = ⊕ d d l1 l1 [3, p. 539]. The subspace l1 consists of all singular functionals on l∞, that is, continuous functionals on l∞ vanishing on compactly supported sequences. In

d particular, l1 contains all extended limits. It is clear from the definition that extended limits form the closed convex ∗ subset of the unit sphere of the space l∞. Hence, by Krein-Milman theorem the set of all extended limits coincide with the closed convex hull of its extreme points. To describe the extreme points of extended limits we shall use another construction of extended limits which is more frequent in set-theoretical expositions. This construction is based on the concept of ultrafilters and we need some definitions to outlined it. In this exposition we mostly follow the book [9].

Definition 2.1.2. A set F of subsets of the set X is said to be a filter on X if the following conditions hold (i) If the set A ⊂ X contains some set from F, then A ∈ F; (ii) The intersection of every finite family of sets from F belongs to F; (iii) F does not contain an empty set.

There are two important examples of filters: - the set of all neighbourhoods of a non-empty set (a point, for example) in a topological space is a filter. Such a filter is called a principal filter; - if X is an infinite set, then the complements of all finite sets form a filter in X. Such a filter is called a Fréchet filter. The following definition introduces the concept of a maximal element in the set of all filters.

14 Definition 2.1.3. A filter F on the set X is said to be an ultrafilter if there is no filter F 0 on X, such that F ( F 0.

It follows directly from the definition, that any principal filter on a non-empty set X is an ultrafilter on X. The following proposition provides a useful characterisation of ultrafilters (see e.g. [9, Chapter I, Proposition 6.4.5 and Proposition 6.4.6]).

Proposition 2.1.4. A ﬁlter F on the set X is an ultraﬁlter if and only if for every A ⊂ X we have that either A or its complement belongs to F.

It follows directly from the latter proposition that Fréchet filter on an infinite set X is not an utrafilter. Indeed, if an infinite set A ⊂ X is such that its complement is also infinite, then neither A nor its complement belongs to a Fréchet filter. Hence, Fréchet filter is not an utrafilter by the later proposition. Another way of looking at ultrafilters on a set X is to define a function f on the power set of X by setting f(A) = 1 if A belongs to an ultrafilter U and f(A) = 0 otherwise. Then f is a finitely additive set function on X. Indeed, let A and B be two disjoint subsets of X. If A ∈ U, then by Definition 2.1.2 (i) and (iii) we have that A∪B ∈ U and B/∈ U. Hence, f(A∪B) = 1 = 1+0 = f(A)+f(B). If A, B∈ / U, then their complements X \ A, X \ B ∈ U. So, by Definition 2.1.2 (ii) we have X \ (A ∪ B) = (X \ A) ∩ (X \ B) ∈ U. Thus, by Definition 2.1.2 (iii), A ∪ B/∈ U and f(A ∪ B) = 0 = f(A) + f(B). For a filter F which is not an ultrafilter the above construction of a function does not work. Indeed, one would say f(A) = 1 if A ∈ F and f(A) = 0 if X \ A ∈ F. However f is left undefined elsewhere. A filter is said to be free if the intersection of all its elements is empty. For example, the Fréchet filter is free. Note that an ultrafilter on an infinite set X is free if and only if it contains the Fréchet filter on X. Indeed, since a free ultrafilter contains no finite set, it follows that it contains all cofinite subsets of X, which is exactly the Fréchet filter. There is an interesting relation between ultrafilters on an infinite discrete X and the Stone-Cech compactification βX of X (see e.g. [3, Section 2.17]). Indeed,

15 it is proved in [3, Theorem 2.86] that there is a homeomorphism between the set βX and the set of all utrafilters on X. Note also, that via this homeomorphism the set X corresponds to principal ultrafilters and set βX \ X corresponds to free ultrafilters.

Deﬁnition 2.1.5. Let f be a mapping from the set X to a topological space Y and let F be a ﬁlter on the set X. An element y ∈ Y is said to be a limit of f

−1 along the ﬁlter F (we write y = limF f) if f (V ) ∈ F for every neighbourhood V of y.

Remark 2.1.6. Let x ∈ l∞ and let F be a filter on N. We consider x as a mapping from N to R. The limit along the filter F equals to some a ∈ R if and only if for every ε > 0 the set {n : |xn − a| < ε} belongs to a filter F.

Note that, the usual limit functional may be considered as a limit along the Fréchet filter. Also, if Y is a Hausdorff space, then every function f : X → Y has at most one limit along any filter [9, Chapter I, Proposition 8.1.1]. If Y is a compact space, then every function f : X → Y has at least one limit along any ultrafilter (see [9, Chapter I, Section 9.1]).

Proposition 2.1.7. The limit along any free utraﬁlter on N is an extended limit on l∞.

Proof. Let U be a free ultraﬁlter on N. Let us show that the functional l : l∞ → R, given by the following formula

l(x) := lim x, x ∈ l∞ U is well-defined. First of all, note that every bounded sequence can be considered N → −k k k k R as a mapping x : [ x l∞ , x l∞ ]. Since the range of x is compact and is Hausdorff, it follows that the limit of x along any utrafilter is unique and the functional l is well-defined. The positivity of l easily follows from Definition 2.1.5. Indeed, for every x ≥ 0 and a < 0 there exists ε = a/2 such that the set {n : |xn − a| < ε} is empty and, thus does not belong to an ultrafilter U. Hence, by Definition 2.1.5 a can not be a limit of x along U.

16 By [9, Chapter I, Proposition 7.3.8] “enlarging” the filter does not change the value of the limit along this filter. Since every free ultrafilter contains the Fréchet filter and since the limit along the Fréchet filter is the usual limit (if exists), it follows that l(x) = limn→∞ xn for every convergent sequence x.

Note that in general the latter proposition does not hold for free filters (since the functional l is not well-defined) and for non-free ultrafilters (since, the limit along non-free ultrafilter does not extend the usual limit).

2.2 Translation invariant extended limits on l∞ (Banach limits)

Let us introduce the extended limits which possess the additional property of translation invariance.

Deﬁnition 2.2.1. A linear functional B on l∞ is said to be a Banach limit if

1. B ≥ 0, that is, Bx ≥ 0 for x ≥ 0,

2. B1I= 1, where 1I= (1, 1,...),

3. BT x = Bx for all x ∈ l∞, where T is a translation operator deﬁned as follows

T (x0, x1,...) = (x1, x2,...).

The existence of Banach limits was established by S. Banach in his treatise [5]. Banach limits proved to be a useful tool in diﬀerent areas, in particular, in noncommutative geometry they are used to construct a Dixmier trace (see e.g. [13,31]). It follows from the deﬁnition that

lim inf xn ≤ Bx ≤ lim sup xn n→∞ n→∞ for every x ∈ l∞ and every Banach limit B. In particular, Bx = limn→∞ xn for

∈ k k ∗ any convergent sequence x l∞ and B l∞ = 1. That is, every Banach limit is

17 an extended limit on l∞. We denote the set of all Banach limits by B. It is clear ∗ that B is the closed convex subset of the unit sphere of the space l∞. Various geometric properties of the set of Banach limits can be found in [82, 85, 86]. In particular, it was established in [82] that the diameter of the set B equals 2. In [86] it was shown that (surprisingly) the radius of the set B equals 2 as well. Moreover, it was proved in [85] that the set B can be split into an uncountable number of disjoint convex subsets with both diameter and radius of each subset equal 2. The following deﬁnition was introduced in [61].

Deﬁnition 2.2.2. A sequence x ∈ l∞ is called almost convergent (to a ∈ R) if Bx = a for every Banach limit B.

The space of all almost convergent sequences is denoted by ac. The following characterisation of almost convergent sequences was proved by G. Lorentz [61].

Theorem 2.2.3. A sequence x ∈ l∞ is almost convergent to a ∈ R if and only if

1 mX+n lim xk = a n→∞ n k=m+1 uniformly in m ∈ N.

Consider a Ces`arooperator C : l∞ → l∞ given by the formula

− 1 Xn 1 (Cx) = x , n ≥ 0. n n k k=0 Clearly, every almost convergent sequence (to a ∈ R) is Ces`aroconvergent, in other words (Cx)n → a. The converse statement does not hold in general (see e.g. Example 2.2.16 below). The latter result was strengthened by L. Sucheston [89] as follows.

Theorem 2.2.4. For every x ∈ l∞ we have

{Bx : B ∈ B} = [q(x), p(x)], where 1 mX+n 1 mX+n q(x) = lim inf x , p(x) = lim sup x . →∞ k →∞ k n m∈N n n m∈N n k=m+1 k=m+1

18 It was shown in [52, 88] that the distance from an element x ∈ l∞ to the subspace ac is calculated by the formula 1 dist(x, ac) = (p(x) − q(x)). 2 We present some examples to illustrate the notion of an almost convergent sequence. Most of them can be found in [85,87].

n Example 2.2.5. A sequence xn = (−1) , n ≥ 0 is almost convergent to zero. Moreover, if x is a periodic sequence and its period is r > 0, then x is almost P 1 r−1 convergent to a number r n=0 xn. The following example provides more sophisticated almost convergent sequences. We shall write Lim x = a if the sequence x is almost convergent to a ∈ R.

Example 2.2.6. The sequence {sin(nt)}n≥0 is almost convergent to zero for every t ∈ R. We have t Xn t t 2 sin( ) sin(jt) = cos( ) − cos(nt + ). 2 2 2 j=1 So, for every t =6 2kπ, k is an integer we have

1 nX+m 1 2 sin(jt) ≤ . n n | sin( t )| j=1+m 2

Hence, by Theorem 2.2.3 the sequence {sin(nt)}n≥0 is almost convergent to zero for every such value of t. If t = 2kπ, then the statement is trivial. Similarly one can show that   1, t = 2kπ, k is integer, { } Lim cos(nt) n≥0 =  0, otherwise.

2 Example 2.2.7. The sequence {sin (nt)}n≥0 is almost convergent for every t ∈ R. 2 1 − Since sin α = 2 (1 cos 2α), it follows from the linearity of Banach limits and 2 the example above that the sequence {sin (nt)}n≥0 is almost convergent and   0, t = kπ, k is integer, { 2 } Lim sin (nt) n≥0 =  1/2, otherwise.

19 Example 2.2.8. The sequence {| sin(nt)|}n≥0 is almost convergent for every t ∈ R. t {| |} If π is a rational number, then the sequence is sin(nt) n≥0 periodic and, hence, almost convergent.

t Let π be an irrational number. Due to periodicity of the sine function one can consider the sequence {nt mod 2π}n≥0 instead of the sequence {nt}n≥0. The sequence {nt mod 2π}n≥0 is uniformly distributed on [0, 2π] (see e.g. [55, Chapter I, §1]). It is proved in [57, Theorem 3.1], that for every uniformly distributed sequence

{xn}n≥0 on [a, b] and every function f continuous on [a, b] the sequence {f(xn)}n≥0 is almost convergent.

So, we conclude that the sequence {| sin(nt)|}n≥0 is almost convergent for every ∈ R t {| |} t such that π is an irrational number. Hence, the sequence sin(nt) n≥0 is Ces`aroconvergent and Lim| sin(nt)| can be evaluated as a Ces`arolimit of the sequence {| sin(nt)|}n≥0. By the classical Weyl Theorem (see e.g. [100] or [6, addition to Chapter XIII,

§26]) for every sequence {xn}n≥0 uniformly distributed on [a, b] and every function f Riemann integrable on [a, b] the sequence {f(xn)}n≥0 is Ces`aroconvergent and

− Z 1 Xn 1 1 b lim f(xn) = f(s)ds. n→∞ n b − a k=0 a So, Z 1 2π 2 lim C{| sin(nt)|} ≥ = | sin(s)|ds = . →∞ n 0 n 2π 0 π Consequently,   Pq  1 sin kπ , t = p π, p, q are integers, q =6 0 | | q q q Lim sin(nt) =  k=1  2 t π , if π is irrational.

A vast array of examples can be obtained considering the operators on l∞ which map the space ac into itself. The following two operators were studied in [98].

Example 2.2.9. Let {mk}k≥0 and nk → ∞ be two sequences of positive integers.

Consider an operator which maps a sequence x = (x0, x1, x2,... ) to a sequence

x = (xm0 , xm0+1, . . . , xm0+n0 ; xm1 , xm1+1, . . . , xm1+n1 ; ... ).

20 It turned out that the sequence x is almost convergent if and only if x is. More- over, Limx = Limx.

Consider now an averaging operator. Fix a sequence {nk}k≥0 and deﬁne an operator P : l∞ → l∞ as follows

n 1 Xk+1 (P x)m = xi, nk ≤ m < nk+1. nk+1 − nk i=nk

It is proved in [98] that the operator P maps ac to itself and also LimP x = Limx.

Let us ﬁnish with the example of the sequence which is not almost convergent.

Example 2.2.10. Consider the Rademacher system given by

n {rn(t) = sign sin(2 πt)}n≥0, t ∈ [0, 1].

According to the law of large numbers q(rn(t)) = −1 and p(rn(t)) = 1 for almost every t ∈ [0, 1] (with respect to Lebesgue measure on [0, 1]). Hence, the sequence

{rn(t)}n≥0 is not almost convergent for almost every t. However, it is clear that the sequence {rn(t)}n≥0 is almost convergent to zero for every dyadic rational t (that is, t = i2−j, 0 ≤ i ≤ 2j, j ∈ N).

As it is shown above both Banach limits and limits along free ultraﬁlters are extended limits on l∞. However, these two classes of extended limits are distinct. The following proposition shows that limits along free ultraﬁlters do not possess the translation invariance property.

Proposition 2.2.11. Let U be a free ultraﬁlter on N. The limit along U is not a Banach limit.

n Proof. Consider the sequence xn = (−1) , n ≥ 0. It is easy to check that every Banach limit vanishes on x. However, the limit along U is either 1 or −1 by Deﬁnition 2.1.5.

It is well-known that translation invariance and multiplicativity of an extended limit are mutually exclusive properties. Indeed, for x = (0, 1, 0, 1,... ) and every Banach limit B we have Bx · BT x = (Bx)2 = 1/4 and B(x · T x) = 0. The following theorem complements Proposition 2.2.11.

21 Theorem 2.2.12. A limit along a free ultraﬁlter is a multiplicative functional on l∞.

Proof. Let U be a free ultraﬁlter on N and let x, y ∈ l∞. Denote a := limU x and b := limU y. Hence, by Deﬁnition 2.1.5 for every ε1, ε2 > 0 sets {n : |xn −a| ≤ ε1} and {n : |yn − b| ≤ ε2} belong to U. ε ε | − | ≤ k k Fix ε > 0 and set ε1 = 2kyk and ε2 = 2kxk . Since xnyn ab y l∞ ε1 + k k x l∞ ε2, it follows that

{n : |xn − a| ≤ ε1} ∩ {n : |yn − b| ≤ ε2} ⊂ {n : |xnyn − ab| ≤ ε} ∈ U, by Deﬁnition 2.1.2 (i) and (ii).

Consequently, limU xy = limU x limU y by Deﬁnition 2.1.5.

If we consider the space l∞ as an algebra, then a multiplicative bounded linear map h : l∞ → C is called a character of l∞. It follows from Theorem 2.2.12 that the set of limits along free ultrafilters is a subset of a positive cone in the set of all characters. We shall describe characters of l∞ (and, so limits along free ultrafilters) from the different points of view.

Let ∆ denotes the set of all characters of l∞. It turns out that characters of l∞ are closely related to maximal ideals of l∞. Indeed, it is proved in [76, Theorem 11] that

- every maximal ideal of l∞ is a kernel of some h ∈ ∆;

- for every h ∈ ∆ the kernel of h is a maximal ideal of l∞. That is, there is one-to-one correspondence between the set of all maximal ideals of l∞ and ∆.

Now we explain the relation between spaces l∞ and ∆. The formula

xˆ(h) := h(x), h ∈ ∆ assigns to each x ∈ l∞ a functionx ˆ : ∆ → C. The functionx ˆ is called a Gelfand transform of x (see e.g. [95, Definition 3.12]). Theorem 11.18 from [76] proves that Gelfand transform is an isometric isomorphism of l∞ onto C(∆). Recall that by βN we denote the Stone-Cech compactification of the space N. Note that βN is a compact Hausdorff space. It is

22 well-known that the space l∞ is isometrically isomorphic to a space C(βN) of all continuous functions on a compact space βN (see e.g. [15, Corollary 15.2]).

Consequently, all three spaces l∞, C(βN) and C(∆) are isometrically isomorphic to one another. It is proved in [76, Example 11.13] that every compact Hausdorﬀ space X can be identiﬁed with the space of maximal ideals of algebra C(X). Hence, it also follows from above that, there is a one-to-one correspondence between the

Stone-Cech compactiﬁcation βN, the space of maximal ideals of algebra l∞ and the set ∆ of all characters of l∞.

In view of the isometric isomorphism between l∞, C(βN) and C(∆) we can combine results from [76, Theorem 11.32], [65] and [3, Theorem 14.12] concerning representations of linear functionals on l∞ to the following theorem.

Theorem 2.2.13. (i) There is a one-to-one correspondence between the set of

all bounded linear functionals on l∞ and the set of all countably additive, regular (in the sense of [65, §3]) set function on βN given by the formula Z l(x) = x(p)dµ(p), βN

where x(p) is a unique extension of x ∈ l∞ to βN.

(ii) The positive functionals l on l∞ correspond to positive set functions µ (that is, a measure) on βN.

(iii) The positive normalised functionals l on l∞ (that is, l(1I)= 1) correspond to a probability measure µ on βN (that is, µ(βN) = 1).

(iii) If K is a subset of a set of all bounded linear functionals on l∞ and M is a corresponding subset of a set of all set function on βN, then extreme points of K correspond to extreme points of M.

It is well-known that the set of extreme points of the set of all extended limits on l∞ consists of multiplicative functionals (see e.g. [62, Theorem 4.12] or [76, Theorem 11.33]). Hence, limits along free utraﬁlters constitute the set of extreme points of extended limits. So, the set of all Banach limits does not contain an extreme point of the set of all extended limits.

23 The structure of the set of extreme points of B is also of interest. Indeed, C. Chou [19] proved that there exists a Banach limit which is not representable as a convex linear combination of countably many extreme points of the set of Banach limits. This result was strengthen in [84], by showing that any Cesàro invariant Banach limit has this property. For the study of the set of extreme points of B we refer a reader to [38,84,96,97]. Nevertheless one can construct Banach limits using ultrafilters. The following well-known proposition simply says that the composition of an extended limit with Cesàrooperator is a Banach limit.

Proposition 2.2.14. The functional

l(x) := lim Cx, x ∈ l∞ U is Banach limit for every free ultraﬁlter U on N.

Proof. Since the C`esarooperator C preserves convergence, it follows from Propo- sition 2.1.7 that l is an extended limit on l∞. The translation invariance of l follows from the “telescoping” property of the C`esarooperator, stating that the sequence C{xn+1 − xn}n≥0 is convergent to zero.

Remark 2.2.15. The construction of Proposition 2.2.14 yields only Banach limits of a special type, called factorisable in [74]. The set of all factorisable Banach limits is a proper subset of the set of all Banach limits. It can be easily seen from the following example.

Example 2.2.16. Let χA denote a characteristic function of a set A ⊂ N and let P ∞ ≥ xn = k=1 χ[2k,2k+k](n), n 0. A direct veriﬁcation shows that the sequence Cx is convergent to zero. So every factorisable Banach limit vanishes on x. However,

we have n 1 mX+n 1 2X+n p(x) = lim sup x ≥ lim x = 1 →∞ k →∞ k n m∈N n n n k=m+1 k=2n+1 and n 1 mX+n 1 2X+2n q(x) = lim inf xk ≤ lim xk = 0. n→∞ m∈N n n→∞ n k=m+1 k=2n+n

Since 0 ≤ xn ≤ 1, it follows that p(x) = 1 and q(x) = 0. Consequently, Theo- rem 2.2.4 yields that {Bx : B ∈ B} = [0, 1].

24 2.3 Extended limits on L∞

We will denote by L∞(0, ∞) and by L∞(R) the spaces of all (classes of) real-valued essentially bounded Lebesgue measurable functions on (0, ∞) and R respectively equipped with the norm

k k | | x L∞ := esssup x(t) , where essential supremum is taken over all t > 0 and t ∈ R respectively. Let

C0(0, ∞) and C0(R) be the spaces of all continuous functions from L∞(0, ∞) and

L∞(R) respectively vanishing at +∞.

Similarly to Deﬁnition 2.1.1 we call a positive linear functional on L∞(R) (or

L∞(0, ∞)) an extended limit if it coincides with the ordinary limit on convergent at +∞ functions.

The following result is well-known. The notation L∞ in the theorem below means either L∞(R) or L∞(0, ∞).

Theorem 2.3.1. For every x ∈ L∞ and for every extended limit γ on L∞ the following inequalities hold

lim inf x(t) ≤ γ(x) ≤ lim sup x(t). t→∞ t→∞ Moreover, the set {γ(x): γ is an extended limit} coincide with the closed interval [lim inft→∞ x(t), lim supt→∞ x(t)].

We shall work with the following linear transformations:

1. Translation operators Tl : L∞(R) → L∞(R), l ∈ R given by

(Tlx)(t) := x(t + l);

2. Dilation operators σ 1 : L∞(0, ∞) → L∞(0, ∞) given by β

(σ 1 x)(t) := x(βt); β

3. Exponentiation operators Pa : L∞(0, ∞) → L∞(0, ∞), a > 0 given by

a (Pax)(t) := x(t );

25 4. The Ces`aromean H : L∞(R) → L∞(R) for the additive group of reals given by Z 1 t (Hx)(t) := x(s) ds; t 0

5. The Ces`aromean M : L∞(0, ∞) → L∞(0, ∞) for the multiplicative group of positive reals given by Z 1 t ds (Mx)(t) := x(s) . log t 1 s

Let us also deﬁne the isomorphism Exp : L∞(0, ∞) → L∞(R) by

(Exp(x)) (t) := (x ◦ exp)(t) := x(et), t ∈ R.

Let Log : L∞(R) → L∞(0, ∞) be an inverse of Exp, that is,

(Log(x)) (t) := (x ◦ log)(t) := x(log t), t > 0.

There are several simple (but important) relations between these transformations, which we will use repeatedly. Some of the proofs below can be also found in [11], where these techniques were pioneered.

Lemma 2.3.2. For every l ∈ R, β > 0, a > 0 we have

Log ◦ Tl ◦ Exp = σe−l ; (2.2)

Log ◦ σa ◦ Exp = Pa; (2.3)

Log ◦ H ◦ Exp = M; (2.4)

HTl − H : L∞(R) → C0(R); (2.5)

Mσ 1 − M : L∞(0, ∞) → C0(0, ∞); (2.6) β

Hσ 1 = σ 1 H; (2.7) β β

MPa = PaM; (2.8)

MH − HM : L∞(0, ∞) → C0(0, ∞). (2.9)

Note, that in (2.7) and (2.9) by H we mean the Ces`arooperator restricted to the space L∞(0, ∞).

26 Proof. We shall prove (2.2). The proofs of (2.3) and (2.4) are similar. For x ∈ L∞(R) we have

t t+l ((Log ◦ Tl ◦ Exp)(x))(t) = (Log(Tlx))(e ) = (x ◦ log)(e )

log t+l l = x(e ) = x(te ) = (σe−l x)(t).

Let us prove (2.5). For x ∈ L∞(R) and every l ∈ R we obtain Z Z t t 1 1 |(HTlx)(t) − (Hx)(t)| = x(s + l)ds − x(s)ds t t Z0 Z 0 l+t t 1 = x(s)ds − x(s)ds |t| Zl Z0 l+t l 1 − = | | x(s)ds x(s)ds t t 0 2|l| ≤ · kxk . |t| L∞

The formula (2.6) follows from (2.5) and (2.4).

For every x ∈ L∞(0, ∞) and β > 0 one has Z Z 1 t 1 βt (Hσ 1 x)(t) = x(βs) ds = x(s) ds = (σ 1 Hx)(t), β β t 0 βt 0 which proves (2.7). The proofs of (2.8) is similar.

The assertion MH − HM : L∞(0, ∞) → C0(0, ∞) is proved in Lemma 5.1.11 below.

We call an extended limit γ on L∞(R) translation invariant if γ ◦ Tl = γ for every l ∈ R. Similarly we define dilation, exponentiation, H- and M-invariant extended limits. As in the preceding sections one can consider ultrafilters on R and show that limits along a free ultrafilter are multiplicative extended limits on L∞(R) or

L∞(0, ∞). It is also clear that there some extended limits (translation invariant, for instance) which are not multiplicative. However, all extended limits “behave multiplicatively” on some functions. The following result is proved in [51]. We shall use it frequently. For the readers convenience we supply a simple proof. The notation L∞ below means either L∞(R) or L∞(0, ∞).

27 Proposition 2.3.3. Let γ be an extended limit on L∞ and let x ∈ L∞ be such that (i) γ(x) = a for some a ∈ R; (ii) either x(t) − a ≤ 0 or else x(t) − a ≥ 0 for suﬃciently large t.

Then γ(xy) = γ(x)γ(y) for every y ∈ L∞.

Proof. Let for simplicity x(t) − a ≥ 0 for t > t0. For every y ∈ L∞ and every t > t0 we have 0 ≤ (x(t) − a)y ≤ kyk(x(t) − a).

7→ − Hence, γ(t (x(t) a)y)χ(t0,∞)) = 0 and

− 7→ − γ((x a)y) = γ(t (x(t) a)yχ(t0,∞)) = 0, that is γ(xy) = γ(x)γ(y).

Another usefull property of extended limits can be now derived from Propo- sition 2.3.3.

Corollary 2.3.4. Let γ be an extended limit on L∞ and let x ∈ L∞ be such that limt→∞ x(t) = a for some a ∈ R. Then γ(xy) = aγ(y) for every y ∈ L∞.

Proof. Since

γ(x − a) = γ(t 7→ (x(t) − a)χ{x≥a}) = γ(t 7→ (x(t) − a)χ{x

γ((x − a)y) = γ(t 7→ (x(t) − a)χ{x≥a}y) + γ(t 7→ (x(t) − a)χ{x

2.4 Translation invariant extended limits on L∞

In this section we discuss translation invariant extended limits on L∞(R). Our main tool is a convex functional deﬁned below in (2.11) is similar, at least in spirit, to L. Sucheston’s functional deﬁned in Theorem 2.2.4. The results given in this section were proved independently in [63] and [77].

28 Deﬁnition 2.4.1. A linear functional ω on L∞(R) is called a translation invariant extended limit if

(i) ω(x) ≥ 0, whenever 0 ≤ x ∈ L∞(R);

(ii) ω(χR) = 1 and ω(χ(−∞,0)) = 0;

(iii) ω(Tlx) = ω(x) for every l ∈ R, x ∈ L∞(R).

Note that in the deﬁnition above the term “extended limit”means “extended limit at +∞”, that is, for every x ∈ L∞(R) such that the limit limt→+∞ x(t) exists we have ω(x) = limt→+∞ x(t). Indeed, if limt→+∞ x(t) = 0, then for every ε > 0 there exists t0 > 0 such that |x(t)| ≤ ε for every t > t0 and using Deﬁnition 2.4.1 (ii) and (iii) we obtain

| | | | | | ≤ ω(x) = ω(xχ(−∞,t0)) + ω(xχ[t0,∞)) = ω(xχ[t0,∞)) ε.

In fact, the main reason why we consider the space L∞(R) instead of L∞(0, ∞) is to describe the crucial relation between translation and dilation invariant extended limits (see Remark 3.1.2 below). In his treatise S. Banach established the existence of extended limits on

L∞(0, ∞), which are invariant under translation operators {Th}h>0 [5, Chapter II, §3]. Using Banach’s approach we sketch the proof of the existence of translation invariant extended limits on L∞(R). Consider the convex functional

1 Xm π(x) := inf lim sup x(t + tk), x ∈ L∞(R), (2.10) t→+∞ m k=1 where inﬁmum is taken over all m ∈ N and all real shifts t1, t2, ..., tm. Applying the Hahn-Banach extension theorem to the functional π, we obtain that there exists a linear functional ω on L∞(R) such that

−π(−x) ≤ ω(x) ≤ π(x), x ∈ L∞(R).

Since π is a positive normalised and translation invariant functional on L∞(R), it follows that ω is a translation invariant extended limit on L∞(R). Moreover, for every function x ∈ L∞(R) the set

{ω(x): ω is a translation invariant extended limit on L∞(R)}

29 coincide with the closed interval [−π(−x), π(x)]. Although the functional π is suitable to prove the existence of translation invariant extended limits, it is given in a quite cumbersome form, which is not very convenient to deal with the extended limits. Our main tool in this section will be a convex “Sucheston-like” functional pT given by the following formula Z 1 t p (x) := lim sup x(s + h)ds, x ∈ L∞(R). (2.11) T → ∞ t + h≥0 t 0

To prove that the functional pT is well-deﬁned we will use the following classical lemma due to M. Fekete [37] (see also [73, I, Problem 98]).

{ }∞ ≤ ∈ N Lemma 2.4.2. Let αn n=1 be such that αn+m αn + αm for every n, m . { }∞ It follows that αn/n n=1 is a convergent sequence and α α lim n = inf n . n→∞ n n∈N n

Lemma 2.4.3. The functional pT is well-deﬁned.

Proof. We shall show that the limit in (2.11) exists.

Fix x ∈ L∞(R) and set Z n αn = sup x(s + h)ds. h≥0 0 For every n, m ∈ N we clearly obtain Z Z n m αn+m = sup x(s + h)ds + x(s + n + h)ds ≤ αn + αm. h≥0 0 0 By Lemma 2.4.2 we have that the sequence

Z ∞ 1 n sup x(s + h)ds ≥ n h 0 0 n=1 converges. Since x ∈ L∞(R), it is straightforward to verify that Z Z 1 t 1 n lim sup x(s + h)ds = lim sup x(s + h)ds. → ∞ →∞ t + t h≥0 0 n n h≥0 0

Lemma 2.4.4. (Properties of the convex functional pT )

(i) pT (x) ≥ 0, whenever x ≥ 0;

30 (ii) pT (χR) = 1 and pT (χ(−∞,0)) = 0;

(iii) pT (Tlx − x) = 0 for every l ∈ R.

Proof. The ﬁrst and the second properties are straightforward consequences of the deﬁnition of pT . For the third property we have Z 1 t p (T x − x) = lim sup (T x(s) − T x(s))ds T l → ∞ l+h h t + h≥0 t 0 = lim sup(HT T x)(t) − (HT x)(t). → ∞ l h h t + h≥0

By (2.5), for every x ∈ L∞(R), t, l ∈ R we have

2|l| |(HT x)(t) − (Hx)(t)| ≤ · kxk . l |t| L∞

Hence,

2|l| 2|l| |p (T x − x)| ≤ lim sup · kT xk = lim · kxk = 0. T l → ∞ h L∞ → ∞ L∞ t + h≥0 |t| t + |t|

The following simple result plays an important role in the proof of Theo- rem 2.4.6.

Lemma 2.4.5. If the map s 7→ hs is the uniformly continuous map from the R R b ∈ R interval [a, b] to the L∞( ), then a hsds L∞( ). Moreover, for every linear functional ω on L∞(R), we have that the function s 7→ ω(hs) is integrable and Z Z b b ω hsds = ω(hs)ds. a a ∈ N k − k ≤ | − | ≤ Proof. Fix ε > 0 and select n such that hs1 hs2 ∞ ε for s1 s2 (b − a)/n. It then follows that

Z − b b − a Xn 1 h ds − h − ≤ (b − a)ε. s n a+i(b a)/n a i=0 P R b−a n−1 ∈ R b Hence, gn = n i=0 ha+i(b−a)/n L∞( ) converges uniformly to a hsds. Therefore,

Z − b b − a Xn 1 ω hsds = lim ω(gn) = lim ω(ha+i(b−a)/n). n→∞ n→∞ n a i=0

31 | − | ≤ k k ∗ k − k Evidently, ω(hs1 ) ω(hs2 ) ω L∞ hs1 hs2 L∞ . Hence, ω(hs) is continuous and, therefore, integrable. Thus,

Z − Z b b − a Xn 1 b ω hsds = lim ω(ha+i(b−a)/n) = ω(hs)ds. n→∞ n a i=0 a

The following theorem was independently proved by A.A. Sedaev, using a somewhat diﬀerent approach [78]. More precisely, he proved that pT (x) = π(x)

(see (2.10)) for every uniformly continuous function from L∞(R), and then applied S. Banach result from [5, Chapter II, §3]. We supply a direct proof.

Theorem 2.4.6. For every uniformly continuous function x ∈ L∞(R) and a ∈ R the following assertions are equivalent:

(1) −pT (−x) ≤ a ≤ pT (x),

(2) There exists a translation invariant extended limit ω on L∞(R) such that ω(x) = a.

Proof. (1)→(2) Let −pT (−x) ≤ a ≤ pT (x). Deﬁne ω on R + xR by the formula

ω(θ1 + θ2x) = θ1 + θ2a.

We have that ω ≤ pT on R + xR. Indeed,

ω(θ1 + θ2x) = θ1 + θ2a ≤ θ1 + |θ2|pT (sgn(θ2)x) = pT (θ1 + θ2x).

By the Hahn-Banach theorem, ω can be extended to L∞(R) preserving the inequality ω ≤ pT . By Lemma 2.4.4 we have

(i) If x ∈ L∞(R) is positive, then 0 ≤ −pT (−x) ≤ ω(x);

(ii) 1 = −pT (−χR) ≤ ω(χR) ≤ pT (χR) = 1 and, similarly, ω(χ(−∞,0)) = 0;

(iii) 0 = −pT (−Tlx + x) ≤ ω(Tlx − x) ≤ pT (Tlx − x) = 0.

So, ω is a translation invariant extended limit on L∞(R) and ω(x) = a.

(2)→(1) Suppose there exists a translation invariant extended limit ω on L∞(R) such that ω(x) = a. We will show that ω(x) ≤ pT (x). Fix t > 0. Since x is uniformly continuous, it follows that the mapping s 7→ Tsx is a uniformly continuous mapping from [0, t] into L∞(R). By Lemma

32 2.4.5, we have Z 1 t ω(x) = ω Tsxds . t 0 ≤ Next, using the fact that ω(x) suph∈R x(h), we obtain Z Z 1 t 1 t ω(x) ≤ sup (Tsx)(h)ds = sup x(s + h)ds. t h∈R 0 t h≥0 0

Letting t → +∞ we obtain ω(x) ≤ pT (x). Substituting x → (−x) to the latter inequality, we obtain −pT (−x) ≤ ω(x).

Deﬁnition 2.4.7. A function x ∈ L∞(R) is said to be almost convergent if all translation invariant extended limits take the same value on x.

The following theorem is an L∞-analogue of the classical Lorentz theorem for l∞ (see Theorem 2.2.3).

Theorem 2.4.8. Let x ∈ L∞(R) be a uniformly continuous function and let a ∈ R. The equality ω(x) = a holds for every translation invariant extended limit

ω on L∞(R) if and only if Z 1 t lim x(s + h)ds = a → ∞ t + t 0 uniformly in h ≥ 0.

Proof. Suppose that all translation invariant extended limits coincide on the uniformly continuous x ∈ L∞(R). Theorem 2.4.6 tells us that for every uniformly continuous function x ∈ L∞(R) the set

{ω(x): ω is a translation invariant extended limit}

coincide with the closed interval [−pT (−x), pT (x)]. Hence, pT (x) = −pT (−x) = a for some a ∈ R. From the deﬁnition of the functional pT we derive, that for every

ε > 0 there exists t0 > 0 such that for all t > t0 we have Z 1 t sup x(s + h)ds < a + ε. h≥0 t 0

Hence Z 1 t x(s + h)ds < a + ε for every h ≥ 0, t > t0. t 0

33 Applying similar arguments to the expression −pT (−x) we obtain that there exists t0 > 0 such that for t > t0 we have Z 1 t a − ε < x(s + h)ds < a + ε for every h ≥ 0. t 0 R 1 t In other words, limt→+∞ x(s + h)ds = a uniformly in h ≥ 0. R t 0 1 t ≥ Let now limt→+∞ t 0 x(s+h)ds = a uniformly in h 0. Then for every ε > 0 there exists t0 > 0 such that for t > t0 we have Z 1 t a − ε < x(s + h)ds < a + ε for every h ≥ 0. t 0 Since h is arbitrary, we infer that Z Z 1 t 1 t a − ε ≤ inf x(s + h)ds ≤ sup x(s + h)ds ≤ a + ε, ∀t > t0. ≥ h 0 t 0 h≥0 t 0

Hence, a − ε ≤ −pT (−x) ≤ pT (x) ≤ a + ε. Since ε is arbitrary, we conclude −pT (−x) = pT (x) = a and so, by Theorem 2.4.6, all translation invariant extended limits coincide on the element x ∈ L∞(R).

At a glance the result of Theorem 2.4.8 is much weaker than that of the original Lorentz theorem 2.2.3 for sequences, since we characterise the almost convergence of uniformly continuous functions only. However, restricting the setting to uniformly continuous functions yields more natural generalisation of Banach limits on sequences to L∞ case. It is easy to see from Lorentz theorem 2.2.3 that every periodic sequence is almost convergent to its average value over the period. It is also clear from Theorem 2.4.8 that every periodic uniformly continuous function x ∈ L∞(R) is almost convergent. The following example (borrowed from [63]) shows that if we drop the assumption of uniform continuity this nice property is not guaranteed anymore.

Example 2.4.9. There exists a periodic non-uniformly continuous function x ∈

L∞(R) such that it is not almost convergent. Let O ⊂ R be a 1-periodic subset (that is, O+1 = O) which is open and dense in R and such that mes(O ∩ [0, 1]) < 1 (here mes means the Lebesque measure on R). Let

x := χO ∈ L∞(R).

34 Note that x is not a continuous function and, moreover, x is not Riemann integrable.

We shall show that π(x) = 1. Fix m ∈ N and real numbers t1, t2, ..., tm. We set \m V := (O − tk). k=1 We shall prove that the set V is non-empty and dense in R. Observe that the set (O − tk) is open and dense in R for every tk ∈ R. Hence, it is suﬃcient to prove that for every A, B ⊂ R such that A and B are both open and dense in R, the set A ∩ B is also open and dense in R. It is easy to see that the set A ∩ B is non-empty and, so, open. We shall show that A ∩ B is dense in R. If t ∈ R belongs either to A or B, then every neighbourhood of t intersects the set A ∩ B, since B (respectively A) is dense in R. Let now t∈ / A ∪ B. Since A is dense in R, it follows that for every ε > 0 the interval (t − ε, t + ε) intersects A. Since A is open, it follows that for every s ∈ A ∩ (t − ε, t + ε) there exists δ > 0 such that (s − δ, s + δ) ⊂ A ∩ (t − ε, t + ε). Next, since B is dense in R, the interval (s − δ, s + δ) contains a point from the set B. Note that this point also belongs to A. Thus, every neighbourhood of t contains a point from A ∩ B, in other words, A ∩ B is dense in R. We have 1 Xm 1 Xm lim sup x(t + tk) ≥ lim sup x(t + tk) = 1, t→+∞ m t→+∞,t∈V m k=1 k=1 since for every t ∈ V we have that t + tk ∈ O for every 1 ≤ k ≤ m and, so x(t + tk) = 1. Hence, 1 Xm π(x) = inf lim sup x(t + tk) ≥ 1. m∈N;t1,t2,...,tm∈R t→+∞ m k=1 ≤ Since π(x) supt∈R x(t) = 1, it follows that π(x) = 1.

It is shown in [63, Example 5.6] that for every 1-periodic function x ∈ L∞(R) we have Z 1 −π(−x) ≤ x(t) dt. 0 So, we have Z 1 −π(−x) ≤ x(t) dt = mes(O ∩ [0, 1]) < 1. 0

35 Hence, −π(−x) < π(x) and the periodic function x is not almost convergent.

It may appear that the extended limits on L∞ invariant under a single operator

Th0 (h0 = 1, for instance) yield a more natural generalisation of Banach limits. Note also that for the construction of Dixmier traces in Theorem 3.2.9 we do not need the extended limit to be invariant under all operators from the translation group {Th}h∈R. It is suﬃcient to consider the invariance under a single operator

Tlog 2 (see proof of Proposition 3.2.11). However, if we consider the class L of all extended limits invariant under a single translation operator T1, then there is a pathological example of a periodic uniformly continuous function, which is not “almost convergent” with respect to the class L. The following example, which is due to A. Sedaev (personal communication), provides an example of such a function.

Example 2.4.10. Let x(t) := cos(2πt), t ∈ R. That is, x is 1-periodic uniformly continuous function.

We claim that there are two extended limits γ1, γ2 ∈ L such that γ1(x) = −1 and γ2(x) = 1, that is, the function x is not “almost convergent” with respect to the class L.

Let us ﬁrst note that if an extended limit is invariant under the operator T1, then it is invariant under {Tn}n∈Z. Consider the functional Xm 0 1 π (x) := inf lim sup x(t + nk), x ∈ L∞(R), (2.12) t→+∞ m k=1 where inﬁmum is taken over all m ∈ N and all non-negative integer shifts n1, n2, ..., nm. Similarly to the original Banach’s proof [5, Chapter II, §3.3] (see also the sketch of the proof at the beginning of this section) one can show that the set L of all extended limits invariant under {Tn}n∈Z is not empty. Moreover, for every function x ∈ L∞(R) the set {l(x): l ∈ L} coincides with the closed interval [−π0(−x), π0(x)].

36 We shall show that if x(t) = cos(2πt), then −π0(−x) = −1 and π0(x) = 1. Indeed,

Xm 0 1 π (x) = inf lim sup x(t + nk) m∈N,n1,...,nm∈Z t→+∞ m k=1 1 Xm ≥ inf lim sup x(t + nk) ∈N ∈Z m ,n1,...,nm t∈Z;t→+∞ m k=1 = 1, since x(k) = 1 for every k ∈ Z. So, π0(x) = 1. Similarly, −π0(−x) = −1.

Hence, there are two extended limits γ1, γ2 ∈ L such that γ1(x) = −1 and

γ2(x) = 1, that is, the function x is not “almost convergent” with respect to the class L.

37 Chapter 3

Dilation invariant extended limits and Dixmier trace construction

In the present chapter we describe the relation between translation and dilation invariant extended limits and construct Dixmier traces using dilation invariant extended limits on L∞(0, ∞). One of the main result of this section is a complete characterisation of Dixmier measurable operators from Mψ (see Deﬁnition 1.1.1). We state this result in terms of both singular values and eigenvalues of an operator.

3.1 Dilation invariant extended limits

Recall that for a given concave function ψ : (0, ∞) → (0, ∞) such that limt→∞ ψ(t) = ∞ and ψ(2t) lim = 1 (3.1) t→∞ ψ(t) and arbitrary translation and dilation invariant extended limit ω on l∞, that is

ω(x0, x1, x2,... ) = ω(x1, x2,... ) and

ω(x0, x0, x1, x1,... ) = ω(x0, x1,... ),

38 J. Dixmier [28] constructed the weight ( )  − 1 Xn 1 Tr (A) := ω  µ(k,A)  , 0 ≤ A ∈ M , (3.2) ω ψ(n) ψ k=0 n≥1 where Mψ is the ideal of compact operators which partial sums of singular values diverge as ψ (see the rigorous deﬁnition in the following section).

This weight extends by linearity to a non-normal trace on Mψ.

To construct the translation and dilation invariant extended limit ω on l∞ J. Dixmier ﬁrst used the fact that there exists a positive linear functional γ on

L∞(R), which is invariant under the group of aﬃne transformations t 7→ at + b on R, a, b ∈ R, a =6 0; and then restricted this functional from L∞(R) to l∞. The latter construction was good enough to establish the existence of singular traces. However, this approach is not so convenient when one deals with singular traces. Hence, many authors use the continuous version of Dixmier’s construction, considering extended limits on L∞(0, ∞) (see e.g. [11,12,21]). In this section following [11] we introduce dilation invariant extended limits on L∞(0, ∞) and describe their correspondence to translation invariant extended limits. Using dilation invariant extended limits we deﬁne the Dixmier traces on a class of Lorentz ideals, which is wider then that of satisfying 3.1. Let us recall that by the dilation operator we mean the operator

σ 1 : L∞(0, ∞) → L∞(0, ∞) β

given by

(σ 1 x)(t) := x(βt), β > 0. β The following deﬁnition should be compared with Deﬁnition 2.4.1

Deﬁnition 3.1.1. A linear functional ω on L∞(0, ∞) is called a dilation invariant extended limit if

(i) ω(x) ≥ 0, whenever 0 ≤ x ∈ L∞(0, ∞);

(ii) ω(χ(0,∞)) = 1 and ω(χ(0,1)) = 0;

(iii) ω(σ 1 x) = ω(x) for every β > 0, x ∈ L∞(0, ∞). β

39 The existence of dilation invariant extended limits easily follows from the existence of the translation invariant extended limits and the following remark (see also [11,29]).

Remark 3.1.2. Let γ be a translation invariant extended limit on L∞(R) (see

Deﬁnition 2.4.1). The functional ω := γ ◦ exp on L∞(0, ∞) deﬁned by setting ω(x) := γ(x ◦ exp) = γ(t 7→ x(et)), is a dilation invariant extended limit on

L∞(0, ∞). Conversely, if ω is a dilation invariant extended limit on L∞(0, ∞), then the functional γ := ω◦log is a translation invariant extended limit on L∞(R).

In view of the previous remark it is natural to consider a “dilation invariant” analogue of the functional pT (see (2.11)). A convex functional pD on L∞(0, ∞) is deﬁned as follows Z 1 t ds pD(x) := pT (x ◦ exp) = lim sup σ 1 x(s) . → ∞ α t + α≥1 log t 1 s The following two theorems describe the set of all dilation invariant extended limits in terms of the functional pD. They are straightforward corollaries (in view of Remark 3.1.2) of Theorems 2.4.6 and 2.4.8. Similar assertions can also be found in [78]. For the convenience we supply short proofs.

Theorem 3.1.3. For every x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous and a ∈ R the following assertions are equivalent:

(1) −pD(−x) ≤ a ≤ pD(x),

(2) There exists a dilation invariant extended limit ω on L∞(0, ∞) such that ω(x) = a.

Proof. In view of the deﬁnition of the functional pD the assertion (1) is equivalent to

−pT (−x ◦ exp) ≤ a ≤ pT (x ◦ exp). (3.3)

Since the function x ◦ exp is uniformly continuous, it follows from Theo- rem 2.4.6 that the condition (3.3) is equivalent to the existence of a translation invariant extended limit γ on L∞(R) such that γ(x ◦ exp) = a. In view of Re- mark 3.1.2 the latter statement is equivalent to the existence of a dilation invariant extended limit ω := γ ◦ exp on L∞(0, ∞) such that ω(x) = a.

40 Theorem 3.1.4. Let x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous and a ∈ R. The equality ω(x) = a holds for every dilation invariant extended limit ω on L∞(0, ∞) if and only if Z 1 t ds lim x(αs) = a → ∞ t + log t 1 s uniformly in α ≥ 1.

Proof. In view of Remark 3.1.2 the equality ω(x) = a holds for every dilation invariant extended limit ω on L∞(0, ∞) if and only if the equality γ(x ◦ exp) = a holds for every translation invariant extended limit γ on L∞(R). Since the function x ◦ exp is uniformly continuous, it follows from Theo- rem 2.4.8 that the latter equality holds if and only if Z 1 t lim (x ◦ exp)(s + h)ds = a → ∞ t + t 0 uniformly in h ≥ 0. By Lemma 2.3.2 the latter expression can be written in the following form Z 1 t ds lim x(αs) = a → ∞ t + log t 1 s uniformly in α ≥ 1, which proves the assertion.

It should be pointed out that the main diﬀerence between Theorems 2.4.6, 2.4.8 and Theorems 3.1.3, 3.1.4 is that the former two theorems are stated for uniformly continuous function x ∈ L∞(R), whereas the latter two holds for the smaller class of functions x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous. As we shall see in Theorem 3.2.13 this smaller class of functions is more than enough for the main purpose of this section, which is to provide a technical base for the study of Dixmier traces.

3.2 Dixmier traces

In this section we present the theory of Dixmier traces on B(H) as it was developed by J. Dixmier, who ﬁrst showed their existence, and then in many subsequent papers including but not limited to [2,29–31,99]. In the following subsection we introduce and discuss the important properties of Lorentz ideals, which are natural domains for Dixmier traces.

41 3.2.1 Lorentz ideals

Recall that (B(H), Tr) denotes the algebra of all bounded linear operators on a Hilbert space H equipped with the uniform norm k · k∞ and a standart trace Tr. Let us recall the notion of generalised singular value function. Given a self- adjoint operator A in H, we denote by EA(·) the spectral measure of A. If A ∈ B(H), then EA(C) ∈ B(H) for all Borel sets C ⊆ R, and there exists s > 0 such that Tr(E|A|(s, ∞)) < ∞. For t ≥ 0, we deﬁne

µ(t, A) = inf{s ≥ 0 : Tr(E|A|(s, ∞)) ≤ t}.

The function µ(A) : [0, ∞) → [0, ∞] is called the generalised singular value function of A; note that µ(t, A) < ∞ for all t > 0.

The latter deﬁnition can be rewritten in terms of the distribution function nA of A. That is, setting

|A| nA(s) := Tr(E (s, ∞)), s ≥ 0, we obtain

µ(t, A) = inf{s : nA(s) ≤ t}, t ≥ 0.

Equivalently, a generalised singular value function µ(A) can be deﬁned by the formula

µ(t, A) = {kApk∞ : p is a projection in B(H) with Tr(1 − p) ≤ t}.

We are interested in the ideal of all compact operators in B(H). Compact operators are “small” operators, that is, if A is a compact operator on H, then for every ε > 0 there exists a finite dimensional subspace E ⊂ H such that kA|E⊥ k < ε. They play the role of infinitesimals in A. Connes quantised calculus [21, Chapter IV] and the size of infinitesimal is determined by the rate of decay of its singular values sequence. For A ∈ B(H) it can be proved that µ(k, A) is the k-th largest eigenvalue of an operator |A| = (A∗A)1/2, k ≥ 0. Moreover, since B(H) is an atomic von Neumann algebra and traces of all atoms equal to 1, P∞ it follows that µ(A) is a step function and µ(A) = k=1 µ(k, A)χ[k−1,k) for every A ∈ B(H).

42 Deﬁnition 3.2.1. By Ω we denote the set of all concave functions ψ : (0, ∞) →

(0, ∞) such that limt→0 ψ(t) = 0 and limt→∞ ψ(t) = ∞.

Now we shall give a rigorous deﬁnition of Lorentz ideals (see e.g. [13,51,58,60]).

∈ M k·k Deﬁnition 3.2.2. Let ψ Ω. A Banach ideal ( ψ, Mψ ) of compact operators in B(H) given by Z t M k k 1 ∞ ψ := A is compact : A Mψ := sup µ(s, A)ds < t>0 ψ(t) 0 is said to be a Lorentz ideal.

For every we set A ∈ Mψ we set Z 1 t a(t, A) := µ(s, A)ds, t > 0. ψ(t) 0

Note that the function a(·,A) depends on ideal Mψ.

In the case when ψ(t) = log(1 + t) the space Mψ is a well-known Dixmier ideal M1,∞.

The norm of Mψ is fully symmetric, in other words,

k k ≤ k k A Mψ B Mψ R R ∈ M t ≤ t for every A, B ψ provided that 0 µ(s, A)ds 0 µ(s, B)ds for every t > 0. Lorentz ideals are an important example of fully symmetric spaces and we refer for a detailed exposition of this theory to [8,54,60]. It is know that Lorentz ideals are not separable. We shall prove this fact in the following subsection using the short argument involving Dixmier traces.

To provide a better description of the ideal M1,∞ we introduce Schatten ideals

Lp, p ≥ 1 (also called noncommutative Lp spaces) as follows n o L k k p 1/p ∞ p := A is compact : A Lp := (Tr(A )) < .

It is clear, that the trace class L1 is a proper subset of all Lorentz ideals M1,∞.

The predual of the ideal M1,∞ is the Macaev ideal (see e.g. [71]) given by Z ∞ µ(t, A) A : dt < ∞ . 0 t + 1

43 By H¨older’sinequality, Macaev ideal contains every Schatten ideal Lq, 1 < q < ∞.

Hence, by duality, Dixmier ideal M1,∞ is contained in every Schatten ideal Lp, p > 1.

Therefore, the ideal M1,∞ is squeezed between Schatten ideals

L1 ( M1,∞ ( Lp, for every p > 1.

3.2.2 Construction of Dixmier traces

In the present subsection we construct non-normal traces via dilation invariant extended limits on L∞(0, ∞) and show that this construction is equivalent to that of J. Dixmier.

Deﬁnition 3.2.3. Let ω be a dilation invariant extended limit on L∞(0, ∞) and let ψ ∈ Ω. If the functional Z 1 t Trω(A) := ω t 7→ µ(s, A)ds , 0 ≤ A ∈ Mψ (3.4) ψ(t) 0 is a weight (that is, a positive homogeneous and additive functional) on the positive cone of Mψ, then the extension of Trω to Mψ by linearity is called a Dixmier trace on Mψ.

The following result, proved in [60] (Theorem 6.3.3), shows, in particular, that not every Lorentz ideal admits non-trivial Dixmier traces.

Theorem 3.2.4. A Lorentz ideal Mψ admits non-trivial Dixmier traces if and only if the function ψ ∈ Ω satisﬁes the following condition

ψ(2t) lim inf = 1. (3.5) t→∞ ψ(t)

∈ 7→ ψ(t) Let us note that due to the concavity of ψ Ω the function t t is decreasing and one has ψ(2t) 1 ≤ ≤ 2, t > 0. ψ(t) 7→ ψ(2t) In particular, the function t ψ(t) is bounded and, so, an extended limit can be applied to it.

44 Deﬁnition 3.2.5. Let ψ ∈ Ω. A dilation invariant extended limit ω on L∞(0, ∞) is said to be ψ-compatible if and only if ψ(2t) ω t 7→ = 1. (3.6) ψ(t)

In view of Theorem 3.2.4 the following result, proved in [51], shows that the existence of ψ-compatible dilation invariant extended limits is equivalent to the existence of non-trivial Dixmier traces.

Theorem 3.2.6. The following are equivalent

(i) The function ψ ∈ Ω satisﬁes (3.5)

(ii) There exists a ψ-compatible dilation invariant extended limit ω on L∞(0, ∞).

∈ ψ(2t) Remark 3.2.7. Note, that when ψ Ω is such that limt→∞ ψ(t) = 1 the existence of ψ-compatible dilation invariant extended limit becomes trivial. Indeed, due to the property of extended limits (see e.g. Theorem 2.3.1) one has that ψ(2t) ω t 7→ = 1 ψ(t) for every dilation invariant extended limit ω on L∞(0, ∞). ψ(2t) In particular, when limt→∞ ψ(t) = 1 every dilation invariant extended limit is ψ-compatible.

The ψ-compatible dilation invariant extended limits on L∞(0, ∞) play a crucial role in the construction of Dixmier traces on Lorentz ideals Mψ. We ﬁrst state an important property of ψ-compatible dilation invariant extended limits. The following lemma is borrowed from [51, Proposition 9].

Lemma 3.2.8. Let ψ ∈ Ω satisfy (3.5) and let ω be a ψ-compatible dilation invariant extended limit on L∞(0, ∞). For every A ∈ Mψ, the function

tµ(t, A) t 7→ , t > 0 ψ(t) belongs to L∞(0, ∞) and tµ(t, A) ω t 7→ = 0. (3.7) ψ(t)

45 Proof. Since the function t 7→ µ(t, A) is non-increasing, it follows that Z t µ(s, A) ds ≥ tµ(t, A), for every t > 0. 0 ∈ M 7→ tµ(t,A) Hence, if A ψ, then the function t ψ(t) is bounded and, so, an extended limit ω can be applied to it. Since ω is a dilation invariant extended limit, it follows that Z Z 1 2t 1 t ω t 7→ µ(s, A) ds = ω t 7→ µ(s, A) ds ψ(2t) ψ(t) 0 0 Z ψ(2t) 1 t = ω t 7→ µ(s, A) ds . ψ(t) ψ(2t) 0

Since ψ is non decreasing, it follows that ψ(2t) ≥ 1 for every t > 0. Since ψ(t) − 7→ ψ(2t) ω is ψ compatible, it follows that ω t ψ(t) = 1 and therefore by Proposi- tion 2.3.3 we have that Z Z ψ(2t) 1 t 1 t ω t 7→ µ(s, A) ds = ω t 7→ µ(s, A) ds . ψ(t) ψ(2t) 0 ψ(2t) 0 Therefore Z Z 1 2t 1 t ω t 7→ µ(s, A) ds = ω t 7→ µ(s, A) ds . ψ(2t) 0 ψ(2t) 0

Hence, Z 1 2t ω t 7→ µ(s, A) ds = 0 ψ(2t) t and, furthermore, 2tµ(2t, A) ω t 7→ = 0. ψ(2t) Again, applying dilation invariance of ω, we arrive at tµ(t, A) ω t 7→ = 0. ψ(t)

The following theorem constructs the singular (Dixmier) traces. It follows the original ideas of J. Dixmier [28] (see also [21,51]). We give its proof here for the completeness.

46 ∈ ψ(2t) Theorem 3.2.9. Let ψ Ω be such that lim inft→∞ ψ(t) = 1. For every ψ- compatible dilation invariant extended limit ω on L∞(0, ∞) the functional Trω deﬁned on the positive cone of Mψ as follows Z 1 t Trω(A) := ω t 7→ µ(s, A)ds ,A ≥ 0 (3.8) ψ(t) 0 extends to a non-normal trace (Dixmier trace) on Mψ.

Proof. Since ω is ψ−compatible, it follows that ψ(2t) ω t 7→ − 1 = 0. ψ(t) ψ(2t) − ≥ Since ψ is non decreasing it follows that ψ(t) 1 0 for every t > 0. By Proposition 2.3.3 we have Z ψ(2t) 1 2t ω t 7→ − 1 µ(s, A) ds = 0, ψ(t) ψ(2t) 0 or, equivalently Z Z 1 2t 1 2t ω t 7→ µ(s, A) ds = ω t 7→ µ(s, A) ds . ψ(2t) 0 ψ(t) 0

Let 0 ≤ A, B ∈ Mψ. We have Z Z Z t/2 t t µ(s, A) + µ(s, B) ds ≤ µ(s, A + B) ds ≤ µ(s, A) + µ(s, B) ds 0 0 0 for every t > 0 (see e.g. [54, Chapter 2] or [60, Theorems 3.3.3 and 3.3.4]). Thus,

Trω(A + B) ≤ Trω(A) + Trω(B).

On the other hand by the remarks above and since ω is dilation invariant, we have Z 1 t Tr (A) + Tr (B) = ω t 7→ µ(s, A) + µ(s, B)ds ω ω ψ(t) Z0 1 2t ≤ ω t 7→ µ(s, A + B)ds ψ(t) Z0 1 2t = ω t 7→ µ(s, A + B)ds ψ(2t) Z 0 1 t = ω t 7→ µ(s, A + B)ds . ψ(t) 0

Thus, we have that Trω(A + B) = Trω(A) + Trω(B) and so Trω extends to a linear functional on Mψ.

47 It follows directly from the construction that any Dixmier trace Trω is a normalised trace on M , that is kTr kM∗ = 1, where k·kM∗ is the norm in the dual ψ ω ψ ψ M∗ M ∗ Banach space ψ = ( ψ) .

As it is mentioned above the ideal Mψ is fully symmetric. There are two important classes of linear functionals on symmetric ideals introduced in the following deﬁnition.

Deﬁnition 3.2.10. A positive linear functional ϕ on Mψ is said to be

- symmetric if ϕ(A) = ϕ(B) whenever 0 ≤ A, B ∈ Mψ are such that µ(A) = µ(B);

-fully symmetric if ϕ(A) ≤ ϕ(B) whenever 0 ≤ A, B ∈ Mψ are such that R R t ≤ t 0 µ(s, A) ds 0 µ(s, B) ds for all t > 0. It is evident that every fully symmetric functional is symmetric. It is also clear from the formula (3.8) that every Dixmier trace is fully symmetric. The example of symmetric functional which fails to be fully symmetric was given in [52]. Next we explain why we call non-normal traces constructed above the “Dixmier traces”, although our construction is diﬀerent from that of J. Dixmier given by (1.1). So, we prove that our “functional” approach is equivalent to Dixmier’s “sequential” one.

∈ ψ(2t) Proposition 3.2.11. Let ψ Ω be such that limt→∞ ψ(t) = 1. The class of traces on Mψ given by the formula (3.8) coincide with the class of Dixmier traces given by (1.1).

Proof. Since the singular value function is a step function, that is µ(t, A) = R P ≤ n n µ(n, A), n t < n + 1, it follows that 0 µ(s, A) ds = k=1 µ(n, A) and that

(original) Dixmier traces are normalised fully symmetric functionals on Mψ.

By [51, Theorem 11] every normalised fully symmetric functional on Mψ is a functional of the form (3.8), constructed via some dilation invariant extended ψ(2t) limit ω on L∞. By Remark 3.2.7 if limt→∞ ψ(t) = 1, then every dilation invariant extended limit is ψ-compatible. Hence, every (original) Dixmier trace is a functional of the form (3.8). We now prove the converse inclusion. In the formula (3.8) the functional ω assumed to be dilation invariant extended limit on L∞, that is, ω = ω◦σβ for every β > 0. In particular, ω = ω◦σ2. Set ω1 :=

48 ω ◦ θ, where θ is an isometric embedding of l∞ into L∞. We conclude that ω1 is an extended limit on l∞, such that ω1(x0, x0, x1, x1,... ) = ω1(x0, x1,... ). Hence, using ω1 one can construct the (original) Dixmier trace. A direct veriﬁcation shows, that ! Z 1 Xn 1 t θ n 7→ µ(k,A) = t 7→ µ(s, A)ds + o(1). ψ(n) ψ(t) k=0 0 Consequently, ! Z 1 Xn 1 t Tr (A) = ω n 7→ µ(k,A) = ω t 7→ µ(s, A)ds . ω1 1 ψ(n) ψ(t) k=0 0 That is, every (original) Dixmier trace is a functional of the form (3.8) and, so, our approach is equivalent to the original Dixmier’s one.

It is easy to see that Dixmier traces vanish on any trace class operator. So, singular traces are very diﬀerent from the classical trace. In fact, the Dixmier traces vanish on the operator ideal which is larger than the trace class ideal.

Indeed, let us consider the closure of the ideal of ﬁnite-rank operators in the Mψ norm, which is the ideal Z 1 t M0 := A ∈ M : lim µ(s, A)ds = 0 . ψ ψ →∞ t ψ(t) 0 M0 It is clear from the construction, that all Dixmier traces vanish on ψ.

Now we shall show that Lorentz ideals Mψ are not separable. It is known that symmetrically normed operator ideal is separable if and only if it is a closure the set of its ﬁnite-rank operators [41, Theorem 3.6.2]. So Mψ is separable if and M M0 { 0 } M only if ψ = ψ. Since Trω(diag ψ (n) n≥1) = 1, we conclude that ψ is not separable.

Recall that an arbitrary A ∈ Mψ has a Jordan decomposition of the form

A = A1 − A2 + iA3 − iA4 , 0 ≤ Aj ∈ Mψ , j = 1, 2, 3, 4.

For convenience, we set

µ˜(A) := µ(A1) − µ(A2) + iµ(A3) − iµ(A4) ,A ∈ Mψ.

49 So for an arbitrary Dixmier trace Trω on Mψ by linearity we have

Trω(A) = Trω(A1) − Trω(A2) + iTrω(A3) − iTrω(A4) Z 1 t = ω t 7→ (µ(s, A ) − µ(s, A ) + iµ(s, A ) − iµ(s, A ))ds ψ(t) 1 2 3 4 Z0 1 t = ω t 7→ µ˜(s, A)ds . ψ(t) 0 Recall the deﬁnition of Dixmier measurable operator, introduced in [21, IV.2.β, Deﬁnition 7]

∈ ψ(2t) ∈ Deﬁnition 3.2.12. Let ψ Ω be such that lim inft→∞ ψ(t) = 1. An operator A

Mψ is called Dixmier measurable if and only if all Dixmier traces Trω coincide on A.

Now we are about to apply the technique developed in previous sections to the problems concerning Dixmier measurability. The following new measurability criteria, which is the main result of this chapter, follows from Theorem 3.1.4.

∈ ψ(2t) ∈ M Theorem 3.2.13. Let ψ Ω be such that limt→∞ ψ(t) = 1. An operator A ψ is Dixmier measurable if and only if the limit Z Z 1 t 1 αs ds lim µ˜(z, A)dz → ∞ t + log t 1 ψ(αs) 0 s exists uniformly in α ≥ 1, moreover Z Z 1 t 1 s ds Tr (A) = lim µ˜(z,A)dz ω → ∞ t + log t 1 ψ(s) 0 s for every Dixmier trace Trω on Mψ. R ∈ M 7→ 1 et Proof. Fix A ψ. We shall ﬁrst check that the function t ψ(et) 0 µ˜(s, A)ds is uniformly continuous. We have ! Z t Z t d 1 e etψ0(et) 1 e etµ˜(et,A) t 7→ µ˜(s, A) ds = − µ˜(s, A) ds + t t t t dt ψ(e ) 0 ψ(e ) ψ(e ) 0 ψ(e )

Z t etψ0(et) 1 e etµ˜(et,A) ≤ µ˜(s, A) ds + . t t t ψ(e ) ψ(e ) 0 ψ(e )

Since Z t 0 t et e ψ (e ) ≤ 1 0 t t ψ (s) ds = 1, ψ(e ) ψ(e ) 0

50 k k k k k k the ﬁrst summand is estimated from above by A1 Mψ + A2 Mψ + A3 Mψ + k k A4 Mψ , where the operators Aj are from the Jordan decomposition of A. For the second summand, since the generalised singular value function is decreasing, it follows that for every B ∈ Mψ we obtain Z t k k 1 ≥ tµ(t, B) B Mψ = sup µ(s, B) ds sup . t>0 ψ(t) 0 t>0 ψ(t) Consequently, ! Z t d 1 e t 7→ µ˜(s, A) ds ≤ 2(kA kM + kA kM + kA kM + kA kM ). t 1 ψ 2 ψ 3 ψ 4 ψ dt ψ(e ) 0 R 7→ 1 et Hence, the function t ψ(et) 0 µ˜(s, A) ds is Lipschitz, so it is uniformly continuous.

By Theorem 3.2.9 an operator A ∈ Mψ is Dixmier measurable if and only if Z 1 t ω t 7→ µ˜(s, A)ds = a ψ(t) 0 for every ψ-compatible dilation invariant extended limit ω and some a ∈ R. ∈ ψ(2t) Since ψ Ω is such that limt→∞ ψ(t) = 1, it follows that ψ(2t) γ t 7→ = 1 ψ(t) for every extended limit γ. In particular, every dilation invariant extended limit on L∞(0, ∞) is ψ-compatible.

Consequently, an operator A ∈ Mψ is Dixmier measurable if and only if Z 1 t ω t 7→ µ˜(s, A)ds = a ψ(t) 0 for every dilation invariant extended limit ω and some a ∈ R. Finally, it follows from Theorem 3.1.4 that the limit Z Z 1 t 1 αs ds lim µ˜(z, A)dz → ∞ t + log t 1 ψ(αs) 0 s exists uniformly in α ≥ 1 and equals to a.

Now we prove the analogue of the classical Lidskii formula for Dixmier traces and restate the measurability criteria of Theorem 3.2.13 in terms of eigenvalues.

51 ∈ ψ(2t) Theorem 3.2.14. Let ψ Ω be such that limt→∞ ψ(t) = 1. For every Dixmier trace Trω on Mψ and every operator A ∈ Mψ the following formula holds   btc 1 X Tr (A) = ω t 7→ λ(k,A) , (3.9) ω ψ(t) k=0 where λ(A) is an eigenvalue sequence of A ordered so that the sequence |λ(A)| is decreasing. Note that this ordering is non necessarily unique. However, the formula above holds for every such ordering.

Proof. First of all, since the singular value function is a step function, it follows that   btc 1 X Tr (A) = ω t 7→ µ˜(k,A) . ω ψ(t) k=0 For every compact operator A there exist a compact normal operator N and a compact quasi-nilpotent operator Q (that is, the spectrum of Q is {0}) such that A = N + Q and σ(A) = σ(N) (multiplicities also coincide) [75, Theorems 1,6,7]. By the Weyl theorem (see e.g. [41, Theorem 3.1]), the sequence |λ(A)| is majorized by the sequence µ(A). So, for A ∈ Mψ, we have N,Q ∈ Mψ. By [60, Theorem

5.5.1] (see also [49]), we have Trω(Q) = 0 for every quasi-nilpotent operator Q and for every Dixmier trace on Mψ. Hence, it is suﬃcient to prove the assertion for normal operators from Mψ. We ﬁrst prove it for self-adjoint operators.

Let A ∈ Mψ be a self-adjoint operator. One clearly have that A = A+ − A−, where A = (|A| A)/2 are positive operators from Mψ. By [60, Lemma 5.2.7] for every compact self-adjoint operator A the following estimate holds

Xn − ≤ ≥ λ(k, A) µ(k, A+) + µ(k, A−) 2(n + 1)µ(n, A), n 0. k=0 Since the function µ(A) is a step function for every A ∈ B(H) and µ(t, A) = µ(n, A) for n ≤ t < n + 1, it follows that

Xbtc

λ(k, A) − µ(k, A+) + µ(k, A−) ≤ 2(t + 1)µ(t, A), t > 0.

k=0 Hence, using Lemma 3.2.8 we obtain that  

Xbtc  1  ω t 7→ λ(k,A) − µ(k, A+) + µ(k, A−) = 0. ψ(t) k=0

52 Consequently, we have     Xbtc Xbtc  1   1  ω t 7→ λ(k,A) = ω t 7→ (µ(k,A ) − µ(k, A−)) . ψ(t) ψ(t) + k=0 k=0

Hence, the assertion is proved for every self-adjoint operator.

For every normal operator N ∈ Mψ we have

Trω(N) = Trω(<(N)) + iTrω(=(N)),

(where <(N) and =(N) are real and imaginary parts of an operator N, respectively). Since operators <(N) and =(N) are self-adjoint, from the ﬁrst part of the proof, it follows that   btc 1 X Tr (N) = ω t 7→ (λ(k, <(N)) + iλ(k, =(N)) . ω ψ(t) k=0

By [60, Lemma 5.2.10] for every compact normal operator N the following estimate holds

Xn − < − = ≤ λ(k, N) λ(k, (N)) iλ(k, (N)) 5(n + 1)µ(n, N). k=0 Again by Lemma 3.2.8, using the same argument as in the self-adjoint case, we obtain   btc 1 X Tr (N) = ω t 7→ (λ(k, <(N)) + iλ(k, =(N)) ω ψ(t)  k=0  btc 1 X = ω t 7→ λ(k,A) . ψ(t) k=0

Using the result of Theorem 3.2.14 we can write the criterion from Theo- rem 3.2.13 in a more computable form. The following theorem provides the Dixmier measurability criterion in terms of eigenvalue sequence of an operator. The proof is similar to that of Theorem 3.2.13.

53 ∈ ψ(2t) ∈ M Theorem 3.2.15. Let ψ Ω be such that limt→∞ ψ(t) = 1. An operator A ψ is Dixmier measurable if and only if the limit

bαkc 1 Xn 1 X lim λ(i, A) n→∞ log(1 + n) kψ(αk) k=1 i=0 exists uniformly in α ≥ 1, moreover

bkc 1 Xn 1 X Trω(A) = lim λ(i, A) n→∞ log(1 + n) kψ(k) k=1 i=0 for every Dixmier trace Trω on Mψ.

Proof. A direct veriﬁcation shows that the function

btc 1 X g(t) := λ(i, A), t > 0 ψ(t) i=0 has jump discontinuities at every point t ∈ N. Since the values of jumps given by λ(btc,A) → ∞ the ratio ψ(t) tending to zero as t , it follows that there exists a function f ∈ L∞(0, ∞) such that f ◦exp is uniformly continuous and f −g vanishes at +∞. R 1 t (As an example of such function f one can take f(t) = ψ(t) 0 θ(λ(A))(s)ds, where

θ is an isometric embedding of l∞ into L∞. As it is proved in Theorem 3.2.10, f ◦exp is uniformly continuous. Also, direct veriﬁcation shows that |f(t)−g(t)| ≤ |λ(btc,A)| → ψ(t) 0.)

By Theorem 3.2.14 the operator A ∈ Mψ is Dixmier measurable if and only if ω(g) = a for every dilation invariant extended limit ω and some a ∈ R. Since f − g vanishes at +∞, it follows that ω(f) = ω(g) for every extended limit ω on L∞(0, ∞). Consequently, an operator A ∈ Mψ is Dixmier measurable if and only if ω(f) = a for every dilation invariant extended limit ω and some a ∈ R. Since f ◦ exp is uniformly continuous, it follows from Theorem 3.1.4 that an operator A ∈ Mψ is Dixmier measurable if and only if the limit Z 1 t ds lim f(αs) (3.10) → ∞ t + log t 1 s exists uniformly in α ≥ 1 and equals a. Since f − g vanishes at +∞, it follows that Z Z t ds t ds f(s) = g(s) + o(log t), t > 0. 1 s 1 s

54 Therefore, the condition (3.10) holds if and only if the limit   Z bαsc 1 t 1 X ds lim  λ(i, A) (3.11) t→+∞ log t ψ(αs) s 1 i=0 exists uniformly in α ≥ 1 and equals a. It is easy to see that the function

bαsc bαkc 1 X 1 X x(s) := λ(i, A) − λ(i, A), k = bsc ψ(αs) ψ(αk) i=0 i=0 tends to zero as s → ∞. R t ds Hence, 1 x(s) s = o(log t) and (3.11) is equivalent to the fact that

bαkc 1 Xn 1 X lim λ(i, A) n→∞ log(1 + n) kψ(αk) k=1 i=0 exists uniformly in α ≥ 1 and equals a.

We now show that the main result of [58] (Theorem 3.7) is a simple corollary of Theorem 3.2.13. It shows that for positive operators the measurability criterion can be written in a simpler form. Although we need to impose the additional technical condition on function ψ ∈ Ω, the result is still useful, since among others the most frequently used ideal M1,∞ (with ψ(t) = log(1 + t)) satisﬁes this condition. This result will be further strengthened in Chapter 4. For the proof of this result we need Hardy’s Tauberian theorem concerning Ces`arosummability (see, e.g. [47, Chapter 6.8]).

Theorem 3.2.16. If x ∈ L∞ is a diﬀerentiable function such that the function t 7→ tx0(t) is bounded from below for suﬃciently large values of t > 0, then the limit Z 1 t lim x(s)ds → ∞ t + t 0 exists if and only if the limit limt→+∞ x(t) exists.

∈ ψ(2t) Corollary 3.2.17. Let ψ Ω be such that limt→∞ ψ(t) = 1 and

d t · log ψ(et) ≤ C (3.12) dt

55 for some C > 0 and every t > 0. An operator 0 ≤ A ∈ Mψ is Dixmier measurable if and only if the limit Z 1 t lim µ(s, A)ds →∞ t ψ(t) 0 exists, moreover Z 1 t Tr (A) = lim µ(s, A)ds ω →∞ t ψ(t) 0 for every Dixmier trace Trω on Mψ.

Proof. Suppose that 0 ≤ A ∈ Mψ is Dixmier measurable. By Theorem 3.2.13 we have that Z Z 1 t 1 αs ds lim µ(z, A)dz → ∞ t + log t 1 ψ(αs) 0 s exists uniformly in α ≥ 1. In particular, Z Z 1 t 1 s ds lim µ(z,A)dz → ∞ t + log t 1 ψ(s) 0 s exists. Making the substitution s 7→ eu we infer that the following limit exists

Z Z Z Z u 1 t 1 s ds 1 t 1 e lim µ(z,A)dz = lim µ(z,A)dz du. → ∞ → ∞ u t + log t 1 ψ(s) 0 s t + t 0 ψ(e ) 0 R 7→ 1 et Let us now check that the function t ψ(et) 0 µ(z,A)dz satisﬁes the Taube- rian condition of Theorem 3.2.16. We have Z ! et · d 7→ 1 t t t µ(z,A)dz dt ψ(e ) 0 Z ! t 0 t et t t · −e ψ (e ) 1 e µ(e ,A) = t t t µ(s, A) ds + t ψ(e ) ψ(e ) 0 ψ(e ) etψ0(et) ≥ − t · kAkM . ψ(et) ψ

The latter expression is bounded, since the function ψ satisﬁes the condition (3.12). Hence, by Theorem 3.2.16 we have that the limit

Z s 1 e lim µ(z,A)dz → ∞ s s + ψ(e ) 0 R 7→ 1 t exists, which clearly implies that the the function t ψ(t) 0 µ(s, A)ds is convergent as t → ∞. The converse implication trivially follows from the Deﬁnition 3.2.3 and the property of extended limits.

56 The following example shows that the result of Corollary 3.2.17 does not necessarily hold for non-positive operators. This example is borrowed from [79, Propositions 8 and 10].

∈ ψ(2t) Example 3.2.18. Let ψ Ω be such that limt→∞ ψ(t) = 1. One can ﬁnd an increasing sequence {tn}n≥0 of integers such that t0 = 0, tk ≥ 2tk−1 and ψ(tk) ≥

2ψ(tk−1) for every k ∈ N. Let a positive operator A be such that

ψ(tk) − ψ(tk−1) µ(t, A) := , tk−1 ≤ t < tk. tk − tk−1 R t ≤ A direct veriﬁcation shows that 0 µ(s, A)ds ψ(t) for every t > 0. Hence,

A ∈ Mψ. Note also, that tkµ(tk,A) ≥ ψ(tk) for every k ∈ N.

Let an operator B be such that µ(B+) = σ2µ(s, A) and µ(B−) = 2µ(s, A).

Clearly, B is a self-adjoint operator, B ∈ Mψ and Trω(B) = 0 for every Dixmier trace Trω, in other words, B is Dixmier measurable. However, for every t > 0 we obtain Z Z 1 t 1 t µ(s, B ) − µ(s, B−)ds = σ µ(s, A) − 2µ(s, A)ds ψ(t) + ψ(t) 2 0 0Z 2 t = − µ(s, A)ds ψ(t) t/2 tµ(t, A) ≤ − , ψ(t) where the last inequality is due to the fact that µ(A) is decreasing. So, Z 1 t tµ(t, A) lim inf µ(s, B ) − µ(s, B−)ds ≤ − lim inf →∞ + →∞ t ψ(t) 0 t ψ(t) t µ(t ,A) ≤ − lim inf n n ≤ −1. →∞ n ψ(tn) R 7→ 1 t Consequently, if the function t ψ(t) 0 µ˜(s, B)ds converges, then Z 1 t lim inf µ˜(s, B)ds ≤ −1, →∞ t ψ(t) 0 which contradicts the last statement of Corollary 3.2.17, since Trω(B) = 0 for every Dixmier trace Trω.

57 We ﬁnish the section constructing a positive operator, which is not Dixmier measurable.

Theorem 3.2.19. The operator A such that

k−ek µ(A) = sup e χ[0,beek c) k≥0 belongs to M1,∞ and it is not Dixmier measurable.

Proof. Consider the function X∞ k−ek k−ek −1 − f := sup e χ[0,eek ) = e χ[eek 1 ,eek ) + e χ[0,e). k≥0 k=1

It is easy to see that µ(A) − f ∈ L1(0, ∞) ∩ L∞(0, ∞). For every n ∈ N and every een ≤ t < een+1 we have Z 1 t a(t, µ(A)) := µ(s, A) ds log(1 + t) 0 ! Z n ee 1 n = f(s)ds + (t − ee )µ(t, A) + O(1) log(1 + t) 0 ! Z k (3.13) Xn ee 1 k = ek−e ds + tµ(t, A) + O(1) log(1 + t) ek−1 k=1 e e en tµ(t, A) = + + o(1). e − 1 log t log(1 + t)

By deﬁnition of an operator A we have that µ(een+1 ,A) = en+2−en+2 . Us- ing (3.13), it is easy to check that e a(t, µ(A)) ≤ + e + o(1), t > 0, e − 1 and, therefore A ∈ M1,∞. However, the limit lim a(t, µ(A)) t→∞ does not exist. Indeed, using (3.13), we obtain

en n+1−en+1 n e e e e lim a(ee , µ(A)) = + = n→∞ e − 1 en e − 1 and, since

en+1 en e + e n n+1 e−n · log = e−n en+1 + log(1 + ee −e ) − log 2 → e, 2

58 it follows that ! een+1 + een e en een+1 + een en+1−en+1 a , µ(A) = + 2 e − 1 een+1 +een 2 een+1 +een log 2 log 2 1 1 en+1 en+1−en(1−e) = + + e − 1 2 een+1 +een een+1 +een log 2 2 log 2 1 1 → + . e − 1 2

Thus, the operator A is not Dixmier measurable by Theorem 3.2.17.

59 Chapter 4

Connes-Dixmier traces

As it was observed by A. Connes [21], in order to ensure that Trω is a positive linear functional on Mψ, it is suﬃcient to only assume that ω is a dilation invariant extended limit on L∞. A natural way to generate such ω’s was suggested by A. Connes [21, Section IV, 2β] by observing that for any generalised limit γ on L∞ a functional ω := γ ◦ M is dilation invariant. Recall that the operator

M : L∞ → L∞ is given by the formula Z 1 t ds (Mx)(t) := x(s) , x ∈ L∞. log t 1 s This class of traces is denoted by C and called Connes-Dixmier traces. A priori, Connes-Dixmier traces form a subset of all Dixmier traces and the question about precise relationship between these two classes arises naturally. One of the main results of this chapter (Theorem 4.1.2) shows that C ( D even on the classical Dixmier ideal M1,∞. ∈ MD ⊂ MC We evidently have that for every ψ Ω the inclusion ψ ψ holds for sets of measurable elements. It was shown in [58, Corollary 3.9] that under some additional condition on the function ψ the classes of positive Dixmier and

Connes-Dixmier measurable operators from Mψ coincide. The second main result of this chapter (Theorem 4.2.3) shows that in general the notions of Dixmier and Connes-Dixmier measurability are distinct even on the positive cone of some

Lorentz space Mψ, ψ ∈ Ω.

60 4.1 The classes of Dixmier and Connes-Dixmier traces are distinct

Throughout this chapter by L∞ we mean L∞(0, ∞), that is, the space of all (classes of) real-valued bounded Lebesgue measurable functions on (0, ∞) equipped with the uniform norm. M0 M Recall that by ψ we denote the separable part of the space ψ, that is, the closure in Mψ of the set of all ﬁnite dimensional operators from B(H). Denote by M0 k − k dist(A, ψ) := inf A B Mψ ∈M0 B ψ ∈ M M0 the distance from an operator A ψ to the subspace ψ. The following lemma was proved in [58, Theorems 2.8 and 5.12] (see also [13, Theorems 7.3 and 7.4]).

Lemma 4.1.1. If ψ ∈ Ω satisﬁes the condition ψ(2t) lim inf = 1, (4.1) t→∞ ψ(t) then M0 ≤ ∈ M dist(A, ψ) = sup Trω(A), 0 A ψ. Trω∈D If ψ satisﬁes the condition d t · log ψ(et) ≤ C (4.2) dt for some C > 0 and every t > 0, then there exists c > 1 such that

≤ M0 ≤ · ≤ ∈ M sup Trω(A) dist(A, ψ) c sup Trω(A), 0 A ψ. Trω∈C Trω∈C This lemma shows that the class of Dixmier traces is large enough to recover the distance of a positive operator from the separable part of Mψ, whereas the class of Connes-Dixmier traces produces the quantity which is equivalent to this distance. In view of this diﬀerence the following question arises naturally: ”Is the constant c in the latter inequality necessarily strictly greater than 1?” The following theorem shows that the inclusion C ⊂ D is proper and answers this question in the aﬃrmative.

61 Theorem 4.1.2. There exists a positive operator A ∈ M1,∞ such that

sup Trω(A) > sup Trω(A). Trω∈D Trω∈C

Proof. Let A be such that

k−ek µ(A) = sup e χ[0,beek c), k≥0 that is, X∞ −1 k−ek − µ(A) = e χ[0,bec) + e χ[bek−1−ek 1 c,beek c). k=1 By Lemma 4.1.1, we have

M0 sup Trω(A) = dist(A, ψ). Trω∈D

It follows from [29, Proposition 2.1] that

M0 dist(A, ψ) = lim sup a(t, µ(A)), t→∞ so

sup Trω(A) = lim sup a(t, µ(A)). Trω∈D t→∞ By the deﬁnition of Connes-Dixmier traces, we have

sup Trω(A) = sup γ(Ma(·, µ(A))), Trω∈C where the supremum is taken over all extended limits γ on L∞. Next, by Theo- rem 2.3.1, we obtain

sup Trω(A) = lim sup(Ma(·, µ(A)))(t). Trω∈C t→∞

So, to prove the assertion it is suﬃcient to show that

lim sup a(t, µ(A)) > lim sup(Ma(·, µ(A)))(t). (4.3) t→∞ t→∞

It follows from Theorem 3.2.19 that A ∈ M1,∞ and X∞ 1 1 n+1 tµ(t, A) a(t, µ(A)) = e χ n n+1 (t) + + o(1), t > 0. (4.4) e − 1 log t [ee ,ee ) log(1 + t) n=1 We ﬁrst evaluate the right-hand side of (4.3).

62 Since for every extended limits γ the functional γ ◦ M is dilation invariant ◦ 7→ tµ(t,A) (see 2.6), it follows from Lemma 3.2.8, that (γ M) t log(1+t) = 0 for every extended limit γ on L∞. Appealing to Theorem 2.3.1, we conclude the function sµ(s, A) M s 7→ (4.5) log(1 + s) vanishes at inﬁnity.

Deﬁne the function x ∈ L∞ by setting ∞ X en x(t) := χ n n+1 (t), t > 0. log t [ee ,ee ) n=0 From (4.4) and (4.5) we obtain e lim sup(Ma(·, µ(A)))(t) = lim sup(Mx)(t). t→∞ e − 1 t→∞ For een ≤ t < een+1 , we have Z 1 t ds (Mx)(t) = x(s) log t s 1Z 1 t ds = x(s) + O(1) log t s e ! − Z ek+1 Z 1 Xn 1 e d log s t d log s = ek + en + O(1) log t ek log s en log s k=0 e !e − 1 Xn 1 = ek + en(log(log t) − n) + o(1) log t k=0 en 1 = + log(log t) − n + o(1). log t e − 1

The function n e 1 n n+1 g : t 7→ + log(log t) − n , t ∈ [ee , ee ) log t e − 1 has extrema at − 1 e1 e−1 +n en en+1 tn = e ∈ [e , e ), n ∈ N.

1 −1 n e−1 ∈ N e 1 We have g(tn) = e for every n . Since g(e ) = lim supt→een+1 − g(t) = e−1 for every n ∈ N , it follows that

1 e e e−1 lim sup(Ma(·, µ(A)))(t) = lim sup(Mx)(t) = . t→∞ e − 1 t→∞ e − 1

63 By the deﬁnition of an operator A we have µ(een ,A) = en+1−en+1 and so, from (4.4) we obtain

sup Trω(A) = lim sup a(t, µ(A)) Trω∈D t→∞ ≥ lim sup a(een , µ(A)) n→∞ n en en (4.4) e e e µ(e , A) = lim sup + − en en n→∞ e 1 log e log!e e een en+1−en+1 = + lim sup n e − 1 n→∞ e e = e − 1 1 e e−1 > e − 1 = lim sup(Ma(·, µ(A)))(t) t→∞

= sup Trω(A). Trω∈C

4.2 The classes of Dixmier and Connes-Dixmier measurable operators are distinct

Although the classes of Dixmier and Connes-Dixmier traces are distinct (The- orem 4.1.2), the corresponding classes of Dixmier and Connes-Dixmier measurable operators from the positive cone of M1,∞ coincide (see e.g. [58, Corollary 3.9]. In this subsection we show that in general it does not hold, that is, there exists a function ψ ∈ Ω such that the notions of Dixmier and Connes-Dixmier measurability diﬀer even on the positive cone of Mψ. The following result is proved in [58] (Corollary 3.9).

Theorem 4.2.1. If ψ ∈ Ω satisﬁes the conditions ψ(2t) lim inf = 1 t→∞ ψ(t) and d t · log ψ(et) ≤ C (4.6) dt

64 for some C > 0 and every t > 0, then for a positive operator A ∈ Mψ the following statements are equivalent: (i) A is Dixmier measurable; (ii) A is Connes-Dixmier measurable;

(iii) There exists Z 1 t lim µ(s, A)ds. →∞ t ψ(t) 0 The theorem below shows that under an additional assumpition on the function ψ the equivalence (i) ⇔ (iii) in the latter theorem holds independently of the condition (4.6).

Theorem 4.2.2. Let ψ ∈ Ω satisfy

ψ(2t) lim = 1. (4.7) t→∞ ψ(t)

A positive operator A ∈ Mψ is Dixmier measurable if and only if the limit Z 1 t lim µ(s, A) ds →∞ t ψ(t) 0 exists.

Proof. Suppose that A ∈ Mψ is a positive Dixmier measurable operator, that is Trω(A) = b for some b ≥ 0 and every Dixmier trace Trω on Mψ. By Theo- rem 3.2.13 we have Z 1 t ds lim a(αs, µ(A)) = b (4.8) →∞ t log t 1 s uniformly in α ≥ 1. Using the pinching theorem one can show that the assumption (4.7) implies

ψ(Nt) lim = 1 for every N > 0. t→∞ ψ(t)

Fix N > 0. One can ﬁnd such t0 = t0(N) that for every t > t0 the following inequality holds ψ(t) 1 ≥ 1 − . (4.9) ψ(Nt) N

By the deﬁnition of a limit superior, there exists α > t0 such that 1 a(α, µ(A)) ≥ 1 − lim sup a(t, µ(A)). (4.10) N t→∞

65 Using (4.9) and (4.10), for every s ∈ [α, αN] we have Z 1 s a(s, µ(A)) = µ(u, A)du ψ(s) 0 Z 1 α ψ(α) ≥ µ(u, A)du = a(α, µ(A)) ψ(αN) ψ(αN) 0 1 2 ≥ 1 − lim sup a(t, µ(A)). N t→∞ Consequently, the substitution s 7→ αs yields Z Z 1 N ds 1 αN ds a(αs, µ(A)) = a(s, µ(A)) log N s log N s 1 α 1 2 ≥ 1 − lim sup a(t, µ(A)). N t→∞ Letting N → ∞ and applying (4.8), we obtain

b ≥ lim sup a(t, µ(A)). t→∞ Similarly one can prove that

b ≤ lim inf a(t, µ(A)) t→∞ and, therefore, lim a(t, µ(A)) = b. t→∞ The converse implication is trivial. √ log(1+t) Let us consider the Lorentz ideal Mψ with ψ(t) = e − 1. It is easy to see that ψ ∈ Ω satisﬁes (4.7). Indeed, √ √ ψ(2t) lim = lim e log(1+2t)− log(1+t) = e0 = 1. t→∞ ψ(t) t→∞

Hence, by Theorem 3.2.4 the ideal Mψ admits non-trivial Dixmier traces. A direct computation shows that √ 1 1 ψ0(t) = e log(1+t) p 2 log(1 + t) 1 + t and d etψ0(et) lim t · log ψ(et) = lim t · →∞ →∞ t t dt t √ψ(e ) log(1+et) t √e p t e = lim t t→∞ e log(1+et) − 1 2 log(1 + et) 1 + e 1 √ = lim t = ∞. 2 t→∞

66 √ Thus, the function ψ(t) = e log(1+t) − 1 does not satisfy (4.6). The following theorem shows that the Dixmier and Connes-Dixmier measurability diﬀer even on the positive cone of the ideal Mψ. √ Theorem 4.2.3. Let ψ(t) = e log(1+t) − 1. There exists a positive Connes-

Dixmier measurable operator A ∈ Mψ such that A is not Dixmier measurable.

Proof. Let A be such that

k−k2 µ(A) = sup e χ[0,bek2 c), k≥0 that is X∞ k−k2 µ(A) = χ[0,1) + e χ[be(k−1)2 c,bek2 c). k=1 Consider the function

k−k2 f := sup e χ[0,ek2 ). k≥0

It is easy to see that µ(A)−f ∈ L1(0, ∞)∩L∞(0, ∞). We obtain for every n ∈ N and every en2 ≤ t < e(n+1)2 ! Z 2 en 1 2 a(t, µ(A)) = µ(s, A)ds + (t − en )µ(t, A) ψ(t) 0 ! Z 2 Xn ek 1 2 = ek−k ds + tµ(t, A) + O(1) (4.11) ψ(t) (k−1)2 k=1 e e en tµ(t, A) = √ + + o(1). e − 1 e log t ψ(t) Now, using (4.11) it is easy to check that

e a(t, µ(A)) ≤ + e + o(1), t > 0, e − 1 and, therefore A ∈ Mψ.

By Lemma 3.2.8, for every dilation invariant extended limit ω on L∞ which is ψ-compatible we have tµ(t, A) ω t 7→ = 0. (4.12) ψ(t) Denote by X∞ √ − n log2 t x(t) := e χ[en2 ,e(n+1)2 )(t). n=0

67 e · We conclude from (4.11) and (4.12) that Trω(A) = e−1 ω(x) for every ψ- compatible dilation invariant extended limit ω on L∞. For every en2 ≤ t < e(n+1)2 , we have ! − Z (k+1)2 Z 1 Xn 1 e √ ds t √ ds (Mx)(t) = ek− log s + en− log s . log t k2 s n2 s k=0 e e √ Making the substitution s → log s and integrating by parts we obtain

Z 2 2 e(k+1) √ √ p e(k+1) k− log s ds k − log s e = e −2e ( log s + 1) k2 s 2 e ek 1 2 = 2 k(1 − ) + 1 − . e e Hence, Z Z 2 t √ e(n+1) √ − ds − ds en log s ≤ en log2 s = o(log t) en2 s en2 s and − 2 Xn 1 1 2 (Mx)(t) = k(1 − ) + 1 − + o(1) log t e e k=0 − 2 e − 1 Xn 1 = k + o(1) log t e k=0 n(n + 1) e − 1 = + o(1) log t e e − 1 = + o(1). e

e−1 In other words, limt→∞(Mx)(t) = e and, therefore, Trω(A) = 1 for every Connes-Dixmier trace Trω. Consequently, A is Connes-Dixmier measurable operator. However, direct computations using the formula (4.11) show that

2 e lim sup a(t, µ(A)) ≥ lim a(en , µ(A)) = t→∞ n→∞ e − 1 and √ 2 e lim inf a(t, µ(A)) ≤ lim a(e(n+1/2) , µ(A)) = . t→∞ n→∞ e − 1 We conclude that the function t 7→ a(t, µ(A)) does not converge at inﬁnity. Hence, by Theorem 4.2.2 A is not Dixmier measurable.

68 Chapter 5

Extended limits with additional invariance properties and Dixmier traces

In the original construction of a non-normal trace (see (1.1)) J. Dixmier used “a linear form invariant under the group of aﬃne transformation t 7→ at+b on R, a, b ∈ R, a =6 0”. Which, of course, means the translation and dilation invariance. The dilation invariance plays a crucial role in the additivity of a functional (1.1). Also, A. Connes in his famous article [20] initially considered precisely this class of ω’s (termed in [20] the ‘class of all means on the amenable group of upper triangular two by two matrices’). Therefore, it is tempting (and natural) to also introduce the class D0 of all Dixmier traces Trω when ω is a translation and dilation invariant extended limit. A priori D0 ⊂ D. One of the main results in this chapter shows that the introduction of the class D0 is, in fact, redundant, since it coincide with the class of all Dixmier traces. We start this chapter considering Ces`aro(H- and M-) invariant extended limits and the class DM of Dixmier traces generated by M-invariant extended limits.

69 5.1 Ces`aroinvariant extended limits

In the present section we consider two Cesàrooperators: the Cesàro(or, Hardy) operator H given by Z 1 t (Hx)(t) := x(s) ds, x ∈ L∞(R) t 0 and the logarithmic Cesàrooperator M given by Z 1 t ds (Mx)(t) := x(s) , x ∈ L∞(0, ∞). log t 1 s We also consider extended limits invariant under the operators H and M. This section organised in the way similar to that of Sections 2.4 and 3.1.

That is, we first define H-invariant extended limits on L∞(R) and describe them in terms of “Sucheston-like” convex functional, after that we define M-invariant extended limits on L∞(0, ∞) and describe the relations between the H- and M- invariant extended limits and finally we discuss the subclass DM of Dixmier traces generated by M-invariant extended limits and DM -measurability. The following definition should be compared with Definition 2.4.1.

Deﬁnition 5.1.1. A linear functional ω on L∞(R) is called an H-invariant extended limit if

(i) ω(x) ≥ 0, whenever 0 ≤ x ∈ L∞(R);

(ii) ω(χR) = 1 and ω(χ(−∞,a)) = 0 for every a ∈ R;

(iii) ω(Hx) = ω(x), for every x ∈ L∞(R).

The existence of H-invariant extended limits was established in [34] (see also [30]). The systematic study of Ces`aroinvariant extended limits on l∞ (that P { } 7→ { 1 n } is, extended limits invariant under the operator xn n≥0 n+1 k=0 xk n≥0, x ∈ l∞) may be found in [83]. Our results extend and partly simplify the technique from [83].

By Lemma 2.3.2 for every l ∈ R, the operator H − HTl maps L∞(R) into

C0, which means that every H-invariant extended limit is translation invariant. Hence, every H-invariant extended limit is an “extended limit at +∞”, which

70 means that if the limit limt→+∞ x(t) exists, then the value of any H-invariant extended limits on x coincide with the limit. So, (H-invariant) extended limits describe the behaviour of a function at +∞, regardless of its behaviour on any interval of the form (−∞, a), a ∈ R.

Note that the main reason to deﬁne H-invariant extended limit on L∞(R) is to describe the relation between H- and M-invariant extended limits (see Re- mark 5.1.10). We ﬁrst prove the following auxiliary result.

Lemma 5.1.2. For every x ∈ L∞(R) we have

lim sup(Hx)(t) ≤ lim sup x(t). t→+∞ t→+∞ k k Proof. The assertion of Lemma 5.1.2 for x + x L∞ and for x are equivalent.

Therefore, we may assume that x ≥ 0. For every ﬁxed t0 > 0 we have

≤ ≤ lim sup(Hx)(t) = lim sup(Hxχ(t0,∞))(t) sup(Hx)(t) sup x(t). t→+∞ t→+∞ t≥t0 t≥t0 → ∞ If t0 + , the right-hand side tends to lim supt→+∞ x(t) and we are done.

Deﬁne a convex positively homogeneous functional pH by the formula ! Xn−1 1 k pH (x) = lim lim sup H x (t), x ∈ L∞(R). (5.1) n→∞ t→+∞ n k=0 The limit in (5.1) exists due to the following lemma.

Lemma 5.1.3. The functional pH is well-deﬁned.

Proof. Fix x ∈ L∞(R) and set ! Xn−1 k αn = lim sup H x (t), n ∈ N. t→+∞ k=0 For every n, m ∈ N we have ! ! ! Xn−1 mX−1 k n k αn+m = lim sup H x (t) + H H x (t) ≤ t→+∞ k=0 k=0 ! ! Xn−1 mX−1 ≤ lim sup Hkx (t) + lim sup Hn Hkx (t). t→+∞ t→+∞ k=0 k=0

71 It follows from Lemma 5.1.2 that

αn+m ≤ αn + αm, ∀n, m ∈ N.

The existence of the limit follows from Lemma 2.4.2.

The following lemma may be compared with Lemma 2.4.4

Lemma 5.1.4. (Properties of the functional pH )

(i) pH (x) ≥ 0, whenever x ≥ 0;

(ii) pH (χR) = 1 and pH (χ(−∞,a)) = 0 for every a ∈ R;

(iii) pH (Hx − x) = 0, for every x ∈ L∞(R).

Proof. The ﬁrst two properties are simple consequences of the deﬁnition of pH . In order to prove the third one, note that

Xn−1 Hk(1 − H) = 1 − Hn. k=0 Therefore, we conclude that k k 1 n 2 x L∞ pH (x − Hx) = lim lim sup (x − H x)(t) ≤ lim = 0. n→∞ t→+∞ n n→∞ n

Similarly, − k k 2 x L∞ pH (x − Hx) ≥ lim = 0. n→∞ n

The following theorem describes the set of all H-invariant extended limits in terms of the functional pH . It is an analogue of L. Sucheston theorem [89] for H-invariant extended limits.

Theorem 5.1.5. For every x ∈ L∞(R) and a ∈ R the following assertions are equivalent:

(1) −pH (−x) ≤ a ≤ pH (x),

(2) There exists an H-invariant extended limit ω on L∞(R) such that ω(x) = a.

72 Proof. (1)→(2). The proof is similar to that of the ﬁrst part of Theorem 2.4.6 and is therefore omitted. (2)→(1). Since ω is an H-invariant extended limit, it follows that for every n ∈ N we obtain ! − 1 Xn 1 a = ω(x) = ω Hkx . n k=0 However, since ω is an extended limit at +∞, it follows that ! ! ! − − − 1 Xn 1 1 Xn 1 1 Xn 1 lim inf Hkx (t) ≤ ω Hkx ≤ lim sup Hkx (t). t→+∞ n n t→+∞ n k=0 k=0 k=0

Letting n → ∞, we conclude −pH (−x) ≤ ω(x) ≤ pH (x).

There are two important corollaries of Theorem 5.1.5.

Corollary 5.1.6. A linear functional ω on L∞(R) is an H-invariant extended limit if and only if ω ≤ pH .

Corollary 5.1.7. For every x ∈ L∞(R) the set

{ω(x): ω is an H-invariant extended limit}

coincide with the closed interval [−pH (−x), pH (x)].

Remark 5.1.8. Although Theorem 5.1.5 is similar to Theorem 2.4.6, there is an important diﬀerence in the settings. Indeed, Theorem 5.1.5 holds for every x ∈

L∞(R), whereas Theorem 2.4.6 holds for uniformly continuous x ∈ L∞(R) only.

Deﬁnition 5.1.9. A linear functional ω on L∞(0, ∞) is called an M-invariant extended limit if

(i) ω(x) ≥ 0, whenever 0 ≤ x ∈ L∞(0, ∞);

(ii) ω(χ(0,∞)) = 1 and ω(χ(0,a)) = 0 for every a > 1;

(iii) ω(Mx) = ω(x) for every x ∈ L∞(0, ∞).

Remark 5.1.10. Let ω and γ be extended limits on L∞(0, ∞) and L∞(R), respectively. Suppose that ω = γ ◦ exp . It follows from the formula (2.4) that ω is M-invariant if and only if γ is H-invariant.

73 Next we prove the following important lemma.

Lemma 5.1.11. Operators (HM − M) and (MH − M) act from L∞(0, ∞) to

C0(0, ∞).

Proof. 1. For every t > 1 we have Z Z Z 1 1 1 t 1 s du (HMx)(t) = (Mx)(s)ds + x(u) ds t t log s u Z0 Z1 Z1 1 1 1 t x(u) t ds = (Mx)(s)ds + du. t 0 t 1 u u log s

We have that Z t ds = li(t) − li(u), u log s where li is a logarithmic integral function deﬁned as follows Z t ds li(t) = . 0 log s For the function li the following asymptotic formula at +∞ holds (see e.g. [1, Chapter 5]) t li(t) = (1 + o(1)). (5.2) log t Hence, we obtain Z Z Z 1 1 li(t) t du 1 t li(u) (HMx)(t) = (Mx)(s)ds + x(u) − x(u) du t t u t u Z0 1 1 Z 1 1 li(t) log t 1 t li(u) = (Mx)(s)ds + · (Mx)(t) − x(u) du. t 0 t t 1 u We clearly have that Z Z t t 1 li(u)du ≤ k k 1 li(u)du x(u) x L∞ . t 1 u t 1 u Integrating the latter term by parts we obtain Z t 1 li(u)du ≤ k k 1 | − | x(u) x L∞ li(t) log t t + 1 t 1 u t

k k li(t) log t − 1 = x L∞ 1 + . t t R Due to (5.2) we conclude that the function t 7→ 1 t x(u) li(u) du belongs to R t 1 u ∞ 7→ 1 1 ∞ C0(0, ). Since the function t t 0 (Mx)(s)ds also belongs to C0(0, ), using (5.2) once more, we obtain that

74 (HMx) ∈ (1 + o(1)) · Mx + C0(0, ∞) for every x ∈ L∞(0, ∞) and therefore the operator HM − M maps L∞(0, ∞) into C0(0, ∞). 2. For every t > 1 we have Z Z 1 t 1 s ds (MHx)(t) = x(u)du log t s s Z1 0 Z Z Z 1 1 t ds 1 t t ds = x(u) du + x(u) du log t s2 log t s2 0 Z 1 Z 1 u Z 1 1 1 1 t x(u) 1 1 t = 1 − x(u)du + du − x(u)du log t t log t u log t t Z0 1 1 1 1 1 1 = 1 − x(u)du + (Mx)(t) − (Hx)(t). log t t 0 log t

Since H is a bounded operator on L∞(0, ∞), it follows that Z 1 1 1 1 (MHx − Mx)(t) = 1 − x(u)du − (Hx)(t) ∈ C0(0, ∞), (5.3) log t t 0 log t for every x ∈ L∞(0, ∞).

We ﬁnish this section with the result establishing that M-invariant extended limits a priori possess some additional invariance properties.

Theorem 5.1.12. Every M-invariant extended limit is necessarily translation, dilation and H-invariant.

Proof. Let ω be M-invariant extended limit and let x ∈ L∞(0, ∞). It follows from Lemma 5.1.11 that

ω(Hx) = ω(MHx) = ω(Mx) = ω(x).

Thus, ω is H-invariant extended limit. Translation and dilation invariance follows from (2.5) and (2.6) respectively.

5.2 On measurability with respect to the sub-

class DM of Dixmier traces

It follows from Theorem 5.1.12 that every M-invariant extended limit is dilation invariant. Hence, for every ψ-compatible M-invariant extended limit ω one

75 can deﬁne the Dixmier trace Trω on Mψ as in Theorem 3.2.9. Denote by DM the class of all such traces.

We clearly have that DM ⊂ D. In this section we show that the later inclusion is proper even on the classical Dixmier-Macaev ideal M1,∞. To this end we ﬁrst prove one technical lemma. Denote by C[1, e] the space of all continuous functions on [1, e]. Deﬁne an auxiliary linear operator L : C[1, e] → L∞(0, ∞) by setting

(Lx)(t) := x(te−blog tc), t > 0, (5.4) that is, X∞ −k (Lx)(t) = x(te )χ[ek,ek+1)(t). k=−∞ Lemma 5.2.1. For every x ∈ C[1, e] we have Z e ds lim (HnLx)(t) = x(s) →∞ n 1 s uniformly in t > 0.

Proof. Deﬁne an operator B : L2(1, e) → L2(1, e) by setting Z Z 1 t 1 e (Bx)(t) = x(s)ds + − x(s)ds, 1 < t < e. t 1 t(e 1) 1 Clearly, Z e 1 1 (Bx)(t) = χ{s

Evidently, Z Z e e |K(s, t)|2 ds dt < ∞, 1 1 that is, B is a Hilbert-Schmidt integral operator and so B is compact operator on L2(1, e). Since B is compact, it follows that every non-zero λ ∈ σ(B) is an eigenvalue. We claim that the only eigenvalue of B is 1 (with the corresponding eigenfunction

χ(1,e)), and so we have σ(B) = {0} ∪ {1}. To see the claim, let us write Z Z t e 1 1 ∈ x(s)ds + − x(s)ds = λx(t), t (1, e). (5.5) t 1 t(e 1) 1

76 Since x ∈ L2(1, e) ⊂ L1(1, e), the left hand side is a continuous function with respect to t. Thus, x ∈ C[1, e]. This implies that the left hand side is continuously diﬀerentiable and so, x ∈ C1[1, e]. Multiplying both sides by t and computing derivatives, we arrive at d x(t) = λ (t 7→ tx(t)), t ∈ [1, e]. dt Hence, x(t) = Ct(1−λ)/λ, t ∈ [1, e]. Substituting this equality in (5.5), we obtain

λ = 1 and x = χ(1,e), which yields the claim. ∗ Let us ﬁnd the operator B . For every x, y ∈ L2(1, e) we have Z e (Bx, y) = (Bx)(t)y(t) dt Z1 Z Z Z e y(t) t e 1 y(t) e = x(s)ds dt + − x(s)ds dt Z1 t Z1 1 (e Z1) t Z1 e e y(t) 1 e e y(t) = x(s) dt ds + − x(s) dt ds. 1 s t (e 1) 1 1 t

Hence, Z Z e e ∗ y(t) 1 y(t) (B y)(s) = dt + − dt. s t (e 1) 1 t −1 Let z(t) = t for t ∈ [1, e] and let Π be the hyperplane in L2(1, e) orthogonal to z. The direct computation yields that B∗z = z. So, for every x ∈ Π we clearly ∗ have (Bx, z) = (x, B z) = (x, z) = 0. Hence, B :Π → Π. Since χ(1,e) ∈/ Π, it n follows that B :Π → Π is quasinilpotent and therefore kB kΠ→Π → 0 as n → ∞. n Thus, B x → 0 in L2(1, e) for every x ∈ Π. We shall now consider B as an operator on the space C[1, e]. It follows from the deﬁnition that B maps positive functions from C[1, e] into positive ones. Therefore, for every x ∈ C[1, e], we have

Bx ≤ kxk∞Bχ[1,e] = kxk∞χ[1,e].

Hence, B :(C[1, e], k · k∞) → (C[1, e], k · k∞) is a contraction. Note that Bx is continuously diﬀerentiable for every x ∈ C[1, e]. Direct computations yields 0 k(Bx) k∞ ≤ (e + 1)kxk∞. Thus, the image of a unit ball (of the space C[1, e]) under B is bounded and equicontinuous. Hence, by Arzel´a-Ascolitheorem [27, Chapter VI, Theorem 3.8] the image of a unit ball under B is totally bounded, that is, B : C[1, e] → C[1, e] is a compact operator.

77 Let x ∈ Π ∩ C[1, e] and let z be a limit point of the sequence Bnx, n ≥ 0.

n n Since B x → 0 in L2(1, e), it follows that z = 0. Hence, B x → 0 in C[1, e] for every x ∈ Π. Next, for every x ∈ C[1, e] we evidently have x − (x, z) ∈ Π. As it was proved above Bχ(1,e) = χ(1,e). Hence, for every x ∈ C[1, e] we obtain

lim (Bnx)(t) = (x, z) + lim (Bn(x − (x, z)))(t) = (x, z) (5.6) n→∞ n→∞ uniformly in t > 0. Let x ∈ C[1, e] and let t > 0. Setting k = blog tc, we obtain ! − Z j+1 Z 1 Xk 1 e t (HLx)(t) = x(se−j)ds + x(se−k)ds t ej ek j=−∞ ! − Z Z −k 1 Xk 1 e te = ej x(s)ds + ek x(s)ds t j=−∞ 1 1 Z Z −k ek e ek te = − x(s)ds + x(s)ds t(e 1) 1 t 1 = (Bx)(te−k) = (Bx)(te−blog tc).

Hence, HL = LB and, therefore, HnLx = LBnx for every n ≥ 1. The assertion follows now from (5.6).

Now we are about to prove the main result of this section. Recall that an operator A ∈ Mψ is measurable with respect to the subclass A of all normalised traces on Mψ if all traces from A take the same value on A. The following theorem shows that even in the classical case of Dixmier-Macaev ideal M1,∞ the set of all Dixmier measurable operators is a proper subset of the set of all DM -measurable operators.

Theorem 5.2.2. The operator A such that

k−ek µ(A) = sup e χ[0,beek c) k≥0 belongs to M1,∞ and it is not Dixmier measurable. However, it is DM -measurable.

Proof. It follows from Theorem 3.2.19 that the operator A ∈ M1,∞ and A is not

Dixmier measurable. It remains to show that A is DM -measurable.

78 It also follows from Theorem 3.2.19 that X∞ 1 1 n+1 tµ(t, A) a(t, µ(A)) = e χ n n+1 (t) + + o(1), t > 0. (5.7) e − 1 log t [ee ,ee ) log(1 + t) n=1 Using the operator L introduced in (5.4), for every dilation invariant extended limit ω on L∞(0, ∞), we obtain e tµ(t, A) Tr (A) = ω (t 7→ a(t, µ(A))) = · ω((Lx) ◦ log) + ω t 7→ , ω e − 1 log(1 + t) where x(t) = t−1 for every t ∈ [1, e). By Lemma 3.2.8 we have, that tµ(t, A) ω t 7→ = 0 log(1 + t) for every dilation invariant extended limit ω on L∞(0, ∞). Hence,

e Tr (A) = (ω ◦ log)(Lx) ω e − 1 for every dilation invariant extended limit ω on L∞(0, ∞). Since every M-invariant extended limit is dilation invariant (see (2.6)), the latter equality also holds for every M-invariant ω. By Remark 5.1.10 we have that if ω is M-invariant, then ω ◦ log is H-invariant. By Lemma 5.2.1, we have Z e ds e − 1 lim (HnLx)(t) = x(s) = →∞ n 1 s e uniformly in t > 0. Therefore,

e − 1 (ω ◦ log)(Lx) = lim (ω ◦ log)(HnLx) = . n→∞ e

Hence, Trω(A) = 1 for every M-invariant extended limit ω, that is, the operator

A is DM -measurable.

As an easy consequence of the latter theorem we have the following corollary.

Corollary 5.2.3. The set DM is a proper subset of the set of all Dixmier traces on M1,∞.

79 5.3 Dilation and Ces`aro-invariant extended limits

In the J. Dixmier’s original construction of a non-normal trace [28], the extended limit ω was assumed to be both translation and dilation (σ2) invariant, whereas many authors assumed only dilation invariance of ω.

Recall that a translation and dilation operators on l∞ are deﬁned as follows

T (x0, x1,...) = (x1, x2,...) and

σ2(x0, x1,...) = (x0, x0, x1, x1,...).

In the present section we explain that these two different approaches yield the same class of traces. Moreover, one can suppose even the stronger condition, that is, both dilation and Cesàroinvariance of an extended limit ω and still obtain the same class of Dixmier traces. First of all, we discuss the existence of such extended limits. Using the Markov-Kakutani fixed point theorem (see e.g. [76, Theorem 5.11]), the existence of a dilation (σ2-only) and Cesàro(and, hence, translation) invariant extended limit on l∞ was shown in [30, Theorem 4.2]. The same proof can be applied to the case of L∞. More recently, N. Kalton constructed such extended limits “explicitly”. We outline this construction. Let B1 and B2 be two Banach limits (on l∞). Consider the functional 7→ k ∈ B(x) := B1 k B2(σ2 x) , x l∞.

It is easy to see that B is a positive normalised linear functional on l∞. Since k 2k k ∈ N clearly σ2 T = T σ2 for every k , it follows that 7→ k 7→ 2k k B(T x) = B1 k B2(σ2 T x) = B1 k B2(T σ2 x) 7→ k = B1 k B2(σ2 x) = B(x),

due to the translation invariance of B2.

80 We also have 7→ k+1 7→ k B(σ2x) = B1 k B2(σ2 x) = B1 T k B2(σ2 x) = B(x),

due to the translation invariance of B1.

Therefore, the functional B is a translation and σ2-invariant extended limit on l∞. If in addition one assumes B2 to be Ces`aroinvariant, then the functional

B is a Ces`aroand σ2-invariant extended limit on l∞.

The similar construction can be applied to extended limits on L∞. Let γ be a translation invariant extended limit on L∞ and let B be a Banach limit on l∞. The functional 7→ 7→ k ∈ ω(x) := B k γ(t σ2 x(t)) , x L∞ is a translation and dilation invariant extended limit on L∞. Similarly if γ is such that γ ◦ H = γ, then the functional ω is a dilation and Cesàroinvariant extended limit on L∞. Finally, note that by Theorem 5.1.12 every M-invariant extended limit yields an example of a dilation and Cesàroinvariant extended limit on L∞. Note that neither this construction nor the proof via Markov-Kakutani fixed point theorem nor the Kalton’s approach do not provide any information concerning the cardinality of the set of all dilation and Cesàroinvariant extended limits. We do this below as a consequence of the following two lemmas.

Lemma 5.3.1. Let x ∈ L∞(0, ∞) be such that x◦exp is uniformly continuous. It follows that ω(x) = ω(Hnx) for every n ∈ N and every dilation invariant extended limit ω on L∞(0, ∞).

Proof. Note that (Hx) ◦ exp is a uniformly continuous function for every x ∈

L∞(0, ∞). Indeed, ! Z t Z t d d 1 e 1 e ◦ − t (Hx exp)(t) = t x(s) ds = t x(s)ds + x(e ), t > 0 dt dt e 0 e 0 and

d 7→ ◦ ≤ k k (t (Hx exp)(t)) 2 x L∞ . dt

81 Since the function x − Hx is such that (x − Hx) ◦ exp uniformly continuous, by Theorem 3.1.3 for every dilation invariant extended limit ω we have

ω(x − Hx) ≤ p (x − Hx) = lim sup((M ◦ σ 1 )(x − Hx))(t). D → ∞ t + α≥1 α

However, by (2.7) the operator σs commutes with H. Therefore, we have

ω(x − Hx) ≤ lim sup((M(1 − H))(σ 1 x))(t). → ∞ t + α≥1 α

By (5.3) we have the following representation

(1 − 1/t)(Hx)(1) − (Hx)(t) ((MH − M)x)(t) = , t > 0. log t k k Since H L∞→L∞ = 1, it follows that

Xn−1 x − Hnx = (1 − H)(Hkx), k=0 it follows that Xn−1 ω(x − Hnx) = ω((1 − H)(Hkx)) = 0. k=0

The following lemma states that the restriction of any dilation invariant extended limit to the set of such functions x ∈ L∞(0, ∞) that x ◦ exp uniformly continuous, possess additional invariance properties.

Lemma 5.3.2. For every dilation invariant extended limit ω on L∞(0, ∞) there exists a dilation and H-invariant extended limit ω0 on L∞(0, ∞) such that ω(x) =

ω0(x) for every x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous.

82 Proof. Let ω be a dilation invariant extended limit on L∞(0, ∞) and let E be the set of all x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous. It follows from Lemma 5.3.1 that ! − 1 Xn 1 ω(x) = ω Hkx . n k=0 Since ω is an extended limit, it follows form Theorem 2.3.1 that ω(x) ≤ ∈ ∞ lim supt→∞ x(t) for every x L∞(0, ). Hence, ! − 1 Xn 1 ω(x) ≤ lim sup Hkx (t) t→∞ n k=0 for every x ∈ E and n ∈ N. Passing n → ∞, we obtain ω(x) ≤ pH (x) for every x ∈ E, where pH is given by the formula (5.1).

By Lemma 5.1.4, pH (x − Hx) = 0 for every x ∈ L∞(0, ∞). Since x − Hx ∈ E for every x ∈ L∞(0, ∞), it follows that

0 = −pH (−x + Hx) ≤ ω(x − Hx) ≤ pH (x − Hx) = 0, ∀x ∈ E.

So, we conclude that ω(x − Hx) = 0, that is, ω is H-invariant on E. Applying invariant form of Hahn-Banach theorem [35, Theorem 3.3.1] to the commutative semigroup

n G = {H σα, n ≥ 0, α ≥ 1}, we extend the restriction of ω on E to the dilation and H-invariant extended limit ω0 on L∞(0, ∞). Finally, by the construction we have that ω|E = ω0|E, that is, ω(x) = ω0(x) for every x ∈ L∞(0, ∞) such that x ◦ exp is uniformly continuous.

Corollary 5.3.3. The cardinality of the set L1 of all dilation and Ces`aro invariant extended limits on L∞ is at least the cardinality of continuum.

Proof. Let x ∈ L∞(0, ∞) be such that x◦exp is uniformly continuous and pD(x) >

−pD(−x). For example, one can choose x(t) := sin(log t) for t > 1 and x(t) := 0 for 0 < t < 1. By Theorem 3.1.3 we have that

{ω(x): ω is dilation invariant extended limit} = [−pD(−x), pD(x)].

83 By Lemma 5.3.2 for every dilation invariant extended limit on L∞(0, ∞) there exists a dilation and H-invariant extended limit ω0 on L∞(0, ∞) such that ω(x) =

ω0(x). Thus,

{ω(x): ω ∈ L1} = [−pD(−x), pD(x)].

Hence, for every a ∈ [−pD(−x), pD(x)] there is ω ∈ L1 such that ω(x) = a.

Consequently, the set L1 contains a subset with the cardinality of continuum.

It follows from the following theorem, that without loss of generality, the extended limit ω used in Deﬁnition 3.2.3 of a Dixmier trace Trω can be assumed simultaneously dilation and H-invariant. Recall, that for every A ∈ Mψ we denote

µ˜(A) = µ(A1) − µ(A2) + iµ(A3) − iµ(A4), where positive operators A1,A2,A3,A4 ∈ Mψ are such that A = A1 − A2 + iA3 − iA4.

Theorem 5.3.4. Let ψ ∈ Ω be such that

ψ(2t) lim inf = 1. t→∞ ψ(t)

Let ω be a ψ-compatible dilation invariant extended limit on L∞(0, ∞). There exists a ψ-compatible dilation and H-invariant extended limit ω0 on L∞(0, ∞) such that Trω = Trω0 .

Proof. As it was established in Theorem 3.2.17 the function Z et 7→ 1 t t µ˜(s, A))ds ψ(e ) 0 is uniformly continuous for every A ∈ Mψ. Hence, by Lemma 5.3.2 there exists a dilation and H-invariant extended limit ω0 on L∞(0, ∞) such that Z Z 1 t 1 t ω0 t 7→ µ˜(s, A))ds = ω t 7→ µ˜(s, A))ds) , (5.8) ψ(t) 0 ψ(t) 0 for every A ∈ Mψ.

84 7→ ψ(2t) ◦ Now we show that ω0 is ψ-compatible. The function t ψ(t) exp is Lipschitz and, therefore, is uniformly continuous. Indeed, t t 0 t t − t t 0 t d 7→ ψ(2e ) 2e ψ (2e )ψ(e ) e ψ(2e )ψ (e ) t t = 2 t dt ψ(e ) ψ (e ) 0 t 0 t t t ψ (2e ) t ψ (e ) ψ(2e ) = 2e − e ψ(2et) ψ(et) ψ(et) ≤ 4, since Z 1 t tψ0(t) 1 = ψ0(s) ds ≥ , t > 0 ψ(t) 0 ψ(t) ψ(2t) ≤ and ψ(t) 2, due to the concavity of ψ. Therefore, by Lemma 5.3.2 ψ(2t) ψ(2t) ω t 7→ = ω t 7→ = 1. 0 ψ(t) ψ(t)

Hence, ω0 is ψ-compatible and the left-hand side of (5.8) coincide with Trω0 (A) for every A ∈ Mψ

Corollary 5.3.5. The classes D and D0 coincide.

85 Chapter 6

Exponentiation invariant extended limits

As it was pointed out in Introduction various important formulae of noncommutative geometry were established for dilation and exponentiation invariant extended limits, where the former assumption was needed in order that the formula (3.8) deﬁnes a Dixmier trace.

In the present chapter we consider the new subclass DP of Dixmier traces on M1,∞ generated by exponentiation invariant extended limits. We prove the analogue of the classical Lidskii formula for the class DP and show that this new traces are, in fact, ζ-function residues. We also study the measurability with respect to the class DP and show that it diﬀers from Dixmier measurability even on the positive cone of M1,∞.

In this chapter we consider Dixmier traces on the ideal M1,∞ only. In the recent paper by V. Gayral and F. Sukochev [40] these constructions were transferred to a general ideal Mψ.

6.1 Dixmier traces generated by exponentiation invariant extended limits

Throughout this chapter by L∞ we mean L∞(0, ∞), that is, the space of all (classes of) real-valued bounded Lebesgue measurable functions on (0, ∞)

86 equipped with the uniform norm. Recall that an exponentiation operator is deﬁned as follows

a (Pax)(t) = x(t ), a > 0, x ∈ L∞.

A extended limit ω on L∞ := L∞(0, ∞) is said to be exponentiation invariant if

ω(Pax) = ω(x) for every x ∈ L∞ and every a > 0.

Remark 6.1.1. For every translation invariant extended limit γ on L∞ the composition γ ◦ exp deﬁnes dilation invariant extended limit (see Remark 3.1.2). Simi- larly, γ ◦exp ◦ exp deﬁnes an exponentiation invariant extended limit. Conversely, if ω is an exponentiation invariant extended limit, then ω ◦ log and ω ◦ log ◦ log are dilation and translation invariant extended limits, respectively.

It follows from [11, Corollary 1.6] (see also [30, Theorem 4.2] and the discussion at the beginning of Section 5.3), that there exists an extended limit γ on L∞ which is both dilation and translation invariant. Hence, by Lemma 2.3.2 the extended limit ω = γ ◦ exp is both dilation and exponentiation invariant. The exact relation between the classes of dilation and exponentiation invariant extended limits remains unknown.

As in Section 3.1, we consider a convex functional pE on L∞(0, ∞) deﬁned by Z 1 t ds p (x) := p (x ◦ exp ◦ exp) = lim sup P x(s) . E T → ∞ a t + a≥1 log log t 0 s log s The following two theorems describe the set of all exponentiation invariant extended limits in terms of the functional pE. They are straightforward corollaries (in view of Remark 6.1.1) of Theorems 2.4.6 and 2.4.8.

Theorem 6.1.2. For every x ∈ L∞(0, ∞) such that x ◦ exp ◦ exp is uniformly continuous and a ∈ R the following assertions are equivalent:

(1) −pE(−x) ≤ a ≤ pE(x),

(2) There exists an exponentiation invariant extended limit ω on L∞(0, ∞) such that ω(x) = a.

87 Theorem 6.1.3. Let x ∈ L∞(0, ∞) such that x◦exp ◦ exp is uniformly continuous and a ∈ R. The equality ω(x) = a holds for every exponentiation invariant extended limit ω on L∞(0, ∞) if and only if Z 1 t ds lim x(sa) = a → ∞ t + log log t 1 s log s uniformly in a ≥ 1.

Note that the condition on the function x ∈ L∞(0, ∞) in Theorems 6.1.2 and 6.1.3 is much stronger than that in Theorems 3.1.3 and 3.1.4. We shall see later in this section that this condition makes the result of Theorem 6.1.3 inapplicable to the study measurability of operators. The following proposition is an analogue of [51, Proposition 10] for exponentiation invariant extended limits. It shows that using Dixmier’s construction with exponentiation invariant extended limit one obtains a non-normal trace.

Theorem 6.1.4. For every exponentiation invariant extended limit ω on L∞ the functional Z 1 t Trω(A) := ω t 7→ µ(s, A) ds , 0 ≤ A ∈ M1,∞, log(1 + t) 0 extends to a non-normal trace on M1,∞.

Proof. Clearly, Trω is positively homogeneous. We only need to prove that Trω is additive on the positive cone of M1,∞.

Let 0 ≤ A, B ∈ M1,∞. By [36, Theorem 4.4 (ii)] we have Z Z Z t/2 t t µ(s, A) + µ(s, B) ds ≤ µ(s, A + B) ds ≤ µ(s, A) + µ(s, B) ds 0 0 0 for every t > 0. Thus, using the positivity of ω, we obtain

Trω(A + B) ≤ Trω(A) + Trω(B) for every extended limit ω. On the other hand, we have Z 1 t Tr (A) + Tr (B) = ω t 7→ µ(s, A) + µ(s, B) ds ω ω log(1 + t) Z0 1 2t ≤ ω t 7→ µ(s, A + B)ds . log(1 + t) 0

88 Since for every ε > 0 we have 2t ≤ t1+ε (for t large enough) and

log(1 + t1+ε) ≤ (1 + ε) log(1 + t),

the following estimates hold Z 1 2t Tr (A) + Tr (B) ≤ ω t 7→ µ(s, A + B)ds ω ω log(1 + t) 0 Z ! t1+ε ≤ 7→ 1 (1 + ε) ω t 1+ε µ(s, A + B)ds . log(1 + t ) 0

Since ω exponentiation invariant, it follows that Z ! t1+ε 7→ 1 ω t 1+ε µ(s, A + B)ds log(1 + t ) 0 Z 1 t = ω t 7→ µ(s, A + B)ds . log(1 + t) 0 Therefore Z 1 t Trω(A) + Trω(B) ≤ (1 + ε) ω t 7→ µ(s, A + B)ds log(1 + t) 0

= (1 + ε) Trω(A + B).

Since ε > 0 can be chosen arbitrary small, we conclude that Trω(A)+Trω(B) ≤

Trω(A + B).

It is obvious from the deﬁnition of Trω in Proposition 6.1.4 that every such functional is fully symmetric (that is, Trω(A) ≤ Trω(B) for every A, B ∈ M1,∞ R R t ≤ t such that 0 µ(s, B)ds 0 µ(s, B)ds, t > 0). By the main result of [51] (Theorem

11) the set of all fully symmetric functionals on M1,∞ coincides with the set of all Dixmier traces, that is, with the set D. Thus, all singular traces generated by the exponentiation invariant extended limits ω are Dixmier traces. Next, we show that the class of Dixmier measurable operators is strictly wider then the class of DP -measurable operators. This fact also proves the inclusion DP ( D.

Deﬁnition 6.1.5. An operator A ∈ M1,∞ is called DP -measurable if Trω(A) takes the same value for all Trω ∈ DP .

89 The criteria for an operator A ∈ M1,∞ (respectively, a positive operator

A ∈ M1,∞) to be Dixmier measurable are given in Theorem 3.2.13 (respectively, Corollary 3.2.17). The crucial point in the proofs of these results is the fact that the function Z et 7→ 1 t t µ(s, A) ds log(1 + e ) 0 is uniformly continuous and, so, one can apply Theorem 3.1.4.

It is desirable to use this approach to study DP -measurability. That is, to apply Theorem 6.1.3, which is an analogue of the Lorentz theorem for exponentiation invariant extended limits. It turned out that we cannot gain any results in this way, since Theorem 6.1.3 requires a function x ∈ L∞(0, ∞) be such that x ◦ exp ◦ exp is uniformly continuous. However, there exists 0 ≤ A ∈ M1,∞ such that the function Z t ee 7→ 1 t et µ(s, A) ds log(1 + e ) 0 is not uniformly continuous. As an example of such A we may take the operator deﬁned in Theorem 6.1.8 below. Hence, we have to use another approach. First, we state two auxiliary lemmas.

Lemma 6.1.6 ( [63, Example 5.6 (1)]). Let x ∈ L∞ be a locally Riemann integrable function. If x is a periodic function and its period is l > 0, then Z 1 l γ(x) = x(s)ds, l 0 for every translation invariant extended limit γ on L∞.

As it was proved in [51, Proposition 9] (see also Lemma 3.2.8) the identity tµ(t, A) ω t 7→ = 0 log(1 + t) holds for every dilation invariant extended limit and for every A ∈ M1,∞. It turned out that the same identity also holds for exponentiation invariant extended limits.

Lemma 6.1.7. For every A ∈ M1,∞ and for every exponentiation invariant extended limit ω, we have tµ(t, A) ω t 7→ = 0. log(1 + t)

90 Proof. Since ω is an exponentiation invariant extended limit, it follows that ! ! Z Z 1+ε t/2 t 7→ 1 7→ 1 2 ω t µ(s, A) ds = ω t 1+ε µ(s, A) ds log(1 + t) 0 log(1 + t ) 0

t1+ε ≥ for every ε > 0. For every ﬁxed ε > 0 we have 2 t (for t large enough) and

log(1 + t1+ε) ≤ (1 + ε) log(1 + t), ∀t > 0.

Hence, Z ! Z 1 t/2 1 1 t ω t 7→ µ(s, A) ds ≥ · ω t 7→ µ(s, A) ds . log(1 + t) 0 1 + ε log(1 + t) 0

Since ε is abritrary, it follows that Z ! Z 1 t/2 1 t ω t 7→ µ(s, A) ds ≥ ω t 7→ µ(s, A) ds . log(1 + t) 0 log(1 + t) 0

The inverse inequality is due to the fact that the singular values function µ(A) is non-negative. Therefore, Z ! Z 1 t/2 1 t ω t 7→ µ(s, A) ds = ω t 7→ µ(s, A) ds log(1 + t) 0 log(1 + t) 0 and Z 1 t ω t 7→ µ(s, A) ds = 0. log(1 + t) t/2 Since µ(A) is decreasing, it follows that Z t tµ(t, A) µ(s, A) ds ≥ , ∀t > 0. t/2 2 Consequently, Z 1 t tµ(t, A) 0 = ω t 7→ µ(s, A) ds ≥ ω t 7→ ≥ 0 log(1 + t) t/2 2 log(1 + t) and tµ(t, A) ω t 7→ = 0. log(1 + t)

The following theorem shows that the sets of all positive DP -measurable and all positive Dixmier measurable operators are distinct.

91 Theorem 6.1.8. There exists a positive Dixmier non-measurable operator A ∈

M1,∞, such that all Trω ∈ DP take the same value on A.

Proof. Let A be a compact operator such that

k−ek µ(A) = sup e χ[0,beek c). k≥0

It follows from Theorem 3.2.19 that the operator A ∈ M1,∞, A is not Dixmier measurable and It follows from Theorem 3.2.19 that A ∈ M1,∞ and X∞ 1 1 n+1 tµ(t, A) a(t, µ(A)) = e χ n n+1 (t) + + o(1), t > 0. (6.1) e − 1 log t [ee ,ee ) log(1 + t) n=1

It remains to show that A is DP -measurable.

By Lemma 6.1.7 for every exponentiation invariant extended limit ω on L∞ we have tµ(t, A) ω t 7→ = 0. (6.2) log(1 + t) Denote ∞ e X ek x(t) := χ k k+1 (t), t > 0. e − 1 log t [ee ,ee ) k=0

We conclude from (6.1) and (6.2) that Trω(A) = ω(x) for every exponentiation invariant extended limit ω on L∞. So, we only need to prove that all exponentiation invariant extended limits coincide on x. By Remark 6.1.1, it is suﬃcient to show that all translation invariant extended limits coincide on x ◦ exp ◦ exp. We clearly have ∞ e X ek (x ◦ exp ◦ exp)(t) = χ (t), e − 1 et [k,k+1) k=0 that is, the function x ◦ exp ◦ exp is a Riemann integrable periodic function. By Lemma 6.1.6, for every translation invariant extended limit γ we have Z e 1 dt e e − 1 γ(x ◦ exp ◦ exp) = = = 1. − t − e 1 0 e e 1 e By Remark 6.1.1 every exponentiation invariant extended limit ω is of the form γ ◦ exp ◦ exp for some translation invariant extended limit γ. So, ω(x) = 1 for every exponentiation invariant extended limit ω. Hence, Z 1 t Trω(A) = ω t 7→ µ(s, A) ds = 1 log(1 + t) 0 for every Trω ∈ DP .

Corollary 6.1.9. The set DP is a proper subset of the set of all Dixmier traces.

92 6.2 Lidskii formula for Dixmier traces generated by exponentiation invariant extended limits

In the present section we ﬁrst prove the analogue of the Lidskii formula for Dixmier traces generated by exponentiation invariant extended limits on the positive cone of M1,∞. Second, using Ringrose’s representation [75, Theorems 1,6,7] of compact operators, we extend the formula to an arbitrary operator A ∈ M1,∞.

Recall that by nA we denote the distribution function of an bounded operator A given by the formula

|A| nA(s) := Tr(E (s, ∞)), s ≥ 0, where EA(·) is a spectral measure of A.

Lemma 6.2.1. For every positive A ∈ M1,∞ and for every exponentiation invariant extended limit ω on L∞ the following formula holds Z ! 1 nA(1/t) Trω(A) = ω t 7→ µ(s, A)ds . log(1 + t) 0

Here, the trace Trω is deﬁned as in Theorem 6.1.4.

Proof. Let A be a positive operator from M1,∞ and ω be an exponentiation invariant extended limit on L∞. From the proof of [11, Proposition 2.4] we know that Z Z t nA(1/t) µ(s, A)ds ≤ µ(s, A)ds + 1. 0 0 Dividing both sides by log(1 + t) and applying ω, we obtain Z ! 1 nA(1/t) Trω(A) ≤ ω t 7→ µ(s, A)ds , t > 0. log(1 + t) 0

On the other hand, by [11, Lemma 2.3] there exists a constant C > 0 such that for t large enough we have

1 n ( ) ≤ C · t log(1 + t) ≤ t1+ε, ε > 0. (6.3) A t

93 We also need the following simple inequality

1 1 + ε 1 + ε = ≤ . (6.4) log(1 + t) log(1 + t)1+ε log(1 + t1+ε)

Hence, by (6.3) and (6.4), using the positivity of µ(A) we obtain ! ! Z Z 1+ε nA(1/t) t 7→ 1 ≤ 7→ 1 + ε ω t µ(s, A)ds ω t 1+ε µ(s, A)ds log(1 + t) 0 log(1 + t ) 0

= (1 + ε)Trω(A), where the latter equality holds since ω is an exponentiation invariant extended limit. Since ε > 0 is arbitrary, we have obtained the converse inequality.

Corollary 6.2.2. If A ∈ M1,∞ is positive, then   1 X Tr (A) = ω t 7→ λ ω log(1 + t) λ∈σ(A),λ>1/t for every Trω ∈ DP .

Proof. Since µ(A) is a step function and since A is selfadjoint, it follows that

Z n (1/t) n (1/t) nA(1/t) AX AX µ(s, A)ds = µ(k, A) + o(1) = λ(k, A) + o(1) 0 k=0 k=0X = λ + o(1), λ∈σ(A):λ>1/t where the latter equality follows from the deﬁnition of a distribution function.

The following corollary is a Lidskii formula for self-adjoint operators from

M1,∞ and Dixmier traces generated by exponentiation invariant extended limits.

∗ Corollary 6.2.3. If A = A ∈ M1,∞, then   1 X Tr (A) = ω t 7→ λ ω log(1 + t) λ∈σ(A),|λ|>1/t for every Trω ∈ DP .

94 Proof. A selfadjoint operator A ∈ M1,∞ can be written as follows A = A+ − A−, with A = (|A|A)/2 be positive operators from Mψ. Note, that for every t > 0 one has X X X X X λ = λ + λ = λ − λ.

λ∈σ(A),|λ|>1/t λ∈σ(A),λ>1/t λ∈σ(A),λ<−1/t λ∈σ(A+),λ>1/t λ∈σ(A−),λ>1/t

The assertion follows from the linearity of Dixmier traces and Corollary 6.2.2.

The following lemma is an analogue of [91, Lemma 42] for exponentiation invariant extended limits.

Lemma 6.2.4. For every positive A ∈ M1,∞ and for every exponentiation invariant extended limit ω on L∞, we have 1 1 ω t 7→ n ( ) = 0. t log(1 + t) A t

Proof. Fix 0 < ε < 1. By the deﬁnition of a distribution function we have 1 1 1 1 X X n ( ) − n ( ) = 1 ≤ λ. t A t A t1−ε t λ∈σ(A):1/t<λ≤1/t1−ε λ∈σ(A):1/t<λ≤1/t1−ε

Dividing both sides by log(1 + t) and applying ω, we obtain 7→ 1 1 − 7→ 1 1 ω t nA( ) ω t nA( 1−ε )  t log(1 + t) t  t log(1 + t) t 1 X ≤ ω t 7→ λ . log(1 + t) 1/t<λ≤1/t1−ε

However, using (6.3) we have

1 1 1 (1−ε)(1+ε) 0 ≤ lim nA( ) ≤ lim t = 0. t→∞ t log(1 + t) t1−ε t→∞ t log(1 + t)

Since ω is a extended limit, it follows that 1 1 ω t 7→ n ( ) = 0. t log(1 + t) A t1−ε

95 Therefore,   1 1 1 X ω t 7→ n ( ) ≤ ω t 7→ λ t log(1 + t) A t log(1 + t) ≤ 1−ε    1/t<λ 1/t  1 X 1 X = ω t 7→ λ − ω t 7→ λ log(1 + t) log(1 + t) 1−ε  λ>1/t   λ>1/t  1 X 1 − ε X = ω t 7→ λ − ω t 7→ λ , log(1 + t) log(1 + t1−ε) λ>1/t λ>1/t1−ε where the last equality holds by Corollary 2.3.4, since

log(1 + t1−ε) lim = 1. t→∞ (1 − ε) log(1 + t)

Using the exponentiation invariance of ω and Corollary 6.2.3, we obtain 1 1 ω t 7→ n ( ) ≤ Tr (A) − (1 − ε)Tr (A) = ε · Tr (A). t log(1 + t) A t ω ω ω

Since 0 < ε < 1 can be chosen arbitrarily small, we conclude that 1 1 ω t 7→ n ( ) = 0. t log(1 + t) A t

We need two auxiliary lemmas to extend the result of Corollary 6.2.3 to non self-adjoint case.

Lemma 6.2.5. For every normal operator A ∈ M1,∞ and for every exponentiation invariant extended limit ω, we have      1 X   1 X  ω t 7→ λ = ω t 7→ λ . log(1 + t) log(1 + t) λ∈σ(A): λ∈σ(A):|<(λ)|>1/t |λ|>1/t or |=(λ)|>1/t

Proof. Consider the diﬀerence

X X X 2 X 2 1 λ − λ ≤ |λ| ≤ 1 ≤ n ( ), t t A t λ∈σ(A): λ∈σ(A):|<(λ)|>1/t λ∈σ(A): λ∈σ(A): |λ|>1/t or |=(λ)|>1/t 1/t≤|λ|≤2/t 1/t≤|λ|≤2/t

96 where the latter inequality follows from the deﬁnition of a distribution function. Hence, using Lemma 6.2.4, we obtain    

1 X  1 X  ω t 7→ λ − ω t 7→ λ log(1 + t) log(1 + t) λ∈σ(A):|λ|>1/t λ∈σ(A):|<(λ)|>1/t   or |=(λ)|>1/t

 1 X X  ≤ ω t 7→ λ − λ  log(1 + t) λ∈σ(A):|λ|>1/t λ∈σ(A):|<(λ)|>1/t or |=(λ)|>1/t 2 1 ≤ ω n ( ) = 0. t log(1 + t) A t

Lemma 6.2.6. For every normal operator A ∈ M1,∞ and for every exponentiation invariant extended limit ω, we have      1 X   1 X  ω t 7→ <(λ) = ω t 7→ <(λ) log(1 + t) log(1 + t) λ∈σ(A):|<(λ)|>1/t λ∈σ(A): or |=(λ)|>1/t |<(λ)|>1/t and      1 X   1 X  ω t 7→ =(λ) = ω t 7→ =(λ) . log(1 + t) log(1 + t) λ∈σ(A):|<(λ)|>1/t λ∈σ(A): or |=(λ)|>1/t |=(λ)|>1/t Proof. Consider the diﬀerence

X X X < − < ≤ |< | (λ) (λ) (λ) λ∈σ(A):|<(λ)|>1/t λ∈σ(A):|<(λ)|>1/t λ∈σ(A):|<(λ)|≤1/t or |=(λ)|>1/t and |=(λ)|>1/t 1 X ≤ 1. t λ∈σ(A):|=(λ)|>1/t

Since the operator A is normal, it follows that =(σ(A)) = σ(=(A)). Hence, we obtain X X 1 1 = 1 = n= ( ) (A) t λ∈σ(A):|=(λ)|>1/t λ∈σ(=(A)):|λ|>1/t and

1 X X 1 1 <(λ) − <(λ) ≤ n= ( ). log(1 + t) t log(1 + t) (A) t λ∈σ(A):|<(λ)|>1/t λ∈σ(A):|<(λ)|>1/t or |=(λ)|>1/t

97 Applying an exponentiation invariant extended limit ω to both sides of the latter expression and using Lemma 6.2.4, we obtain the required equality. The proof of the second equality is similar and is therefore omitted.

The following theorem is an analogue of Lidskii formula for Dixmier traces generated by exponentiation invariant extended limits.

Theorem 6.2.7. For every operator A ∈ M1,∞ and for every exponentiation invariant extended limit ω, we have   1 X Tr (A) = ω t 7→ λ . ω log(1 + t) λ∈σ(A):|λ|>1/t

Proof. By [75, Theorems 1,6,7], for every compact operator A there exist a compact normal operator S and a compact quasi-nilpotent operator Q such that A = S + Q and σ(A) = σ(S) (multiplicities coincide as well). By Weyl theorem (see e.g. [41, Theorem 3.1]), the sequence |λ(A)| is majorized by the sequence µ(A). So, for A ∈ M1,∞, we have S, Q ∈ M1,∞. By [49, Theorem

3.3], we have Trω(Q) = 0 for every quasi-nilpotent operator Q and for every trace. Hence, it is suﬃcient to prove the statement of the theorem for a normal operator

A ∈ M1,∞.

Let A ∈ M1,∞ be normal and let ω be an exponentiation invariant extended limit. Since A is normal, it follows that <(σ(A)) = σ(<(A)) and X X <(λ) = λ. λ∈σ(A):|<(λ)|>1/t λ∈σ(<(A)):|λ|>1/t

Then by Corollary 6.2.3 we obtain   1 X Tr (<(A)) = ω t 7→ <(λ) . ω log(1 + t) λ∈σ(A):|<(λ)|>1/t

Similarly, for the operator =(A) we obtain   1 X Tr (=(A)) = ω t 7→ =(λ) . ω log(1 + t) λ∈σ(A):|=(λ)|>1/t

The assertion of the theorem follows from Lemma 6.2.6 and Lemma 6.2.5.

98 6.3 Zeta-function residues and Dixmier traces

In this section we describe the relation between Dixmier traces generated by exponentiation invariant extended limits and ζ-functions of non-commutative geometry. Let A be a positive operator from B(H). Consider the following function 1 t 7→ Tr A1+1/t , t > 0. (6.5) t When this function is bounded it is frequently referred to as ζ-function associated with the operator A. It follows from [12, Theorem 4.5] that the function (6.5) is bounded if and only if A ∈ M1,∞. The asymptotic behaviour of ζ-functions at +∞ is of a particular interest. This asymptotic can be expressed with the help of some extended limit γ on L∞(0, ∞) yielding the functional 1 1+1/t ζ (A) := γ t 7→ Tr A , 0 ≤ A ∈ M ∞ (6.6) γ t 1, called a ζ-function residue. Observe that, in the construction of ζ-function residues, we do not restrict our attention to dilation (or exponentiation) invariant extended limits. It follows from [94, Theorem 8] that for every extended limit γ on L∞ the functional (6.6) extends by linearity to a fully symmetric functional on M1,∞. Hence, by [51, Theorem 11] every ζ-function residue coincides with some Dixmier trace. The following result speciﬁes this correspondence between ζ-function residues and Dixmier traces, although for a particular type of extended limits.

Theorem 6.3.1 ( [94, Theorem 15]). If ω is a dilation and exponentiation invariant extended limit, then

Trω = ζω◦log.

The latter theorem was ﬁrst proved in [11] under even more restrictive conditions on ω, that are dilation, exponentiation and M-invariance. Although, the class of all Dixmier traces is strictly larger than that of all ζ- function residues (see [60, Theorem 8.7.1]), it was shown in [14, Theorem 7] that if all ζ-function residues coincide on a positive operator A ∈ M1,∞, then A is Dixmier measurable.

99 To extend Theorem 6.3.1 by dropping the condition of dilation invariance we need the following auxiliary theorem proved in [11] (Theorem 2.2).

Theorem 6.3.2 (weak∗-Karamata theorem). Let ω be a dilation invariant extended limit on L∞ and let β be a real valued, increasing, right continuous function on (0, ∞) such that the integral

Z ∞ − t h(r) = e r dβ(t) 0 converges for all r > 0. Then h(r) β(t) ω r 7→ = ω t 7→ . r t

The following theorem extends the result of Theorem 6.3.1 by weakening the assumptions on an extended limit ω.

Theorem 6.3.3. If ω is an exponentially invariant extended limit, then

Trω = ζω◦log.

Proof. It is suﬃcient to prove the equality Trω = ζω◦log on the positive cone of

M1,∞. Without loss of a generality kAk∞ ≤ 1. Fix A ∈ M1,∞. Deﬁne the function β : (0, ∞) → (0, ∞) by Z t −s −s β(t) := − e dnA(e ). 0 Since the change of variable λ = e−s is strictly decreasing function, the function β is non-negative and increasing. Let

Z ∞ − t h(r) := e r dβ(t). 0

−s s It follows from (6.3) that nA(e ) ≤ Cse for large values of s > 0. Hence, the integral h(r) converges for all r > 0. So, β satisﬁes the assertions of Theorem 6.3.2.

Since kAk∞ ≤ 1, it follows from the spectral theorem (see e.g. [64, Deﬁnition

15.1.1]) that Z 1 1+1/r 1+1/r Tr(A ) = λ dnA(λ). 0 Setting λ = e−t, we obtain

100 Z ∞ Z ∞ 1+1/r −t−t/r −t − t Tr(A ) = e dnA(e ) = e r dβ(t) = h(r) 0 0 and by the deﬁnition of ζ-function we have 1 1+1/r h(r) ζ ◦ (A) = (ω ◦ log) r 7→ Tr(A ) = (ω ◦ log) r 7→ . ω log r r

By Remark 6.1.1 the functional ω ◦ log is a dilation invariant extended limit. Hence by Theorem 6.3.2 h(r) β(t) (ω ◦ log) r 7→ = (ω ◦ log) t 7→ . r t

Thus we obtain β(t) ζ ◦ (A) = (ω ◦ log) t 7→ ω log t Z 1 0 = (ω ◦ log) t 7→ − e−s dn (e−s) t A Z t 1 1 = (ω ◦ log) t 7→ s dnA(s) t −t Z e 1 1 = ω t 7→ s dnA(s) . log t 1/t

By the deﬁnition of a distribution function nA we have Z Z 1/t nA(1/t) − s dnA(s) = µ(s, A)ds. 1 0

Hence, Z ! 1 nA(1/t) ζω◦log(A) = ω t 7→ µ(s, A)ds = Trω(A), log t 0 where the last equality is due to Theorem 6.2.1.

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