Singular Traces: Theory and Applications 9783110262551

Table of contents :
Preface
Introduction
I Preliminary Material
1 What is a Singular Trace?
1.1 Compact Operators
1.2 Calkin Correspondence
1.3 Examples of Traces
1.3.1 The Canonical Trace
1.3.2 The Dixmier Trace
1.3.3 Lidskii Formulation of Traces
1.4 Notes
2 Preliminaries on Symmetric Operator Spaces
2.1 Von Neumann Algebras
2.2 Semifinite Normal Traces
2.3 Generalized Singular Value Function
2.4 Calkin Correspondence in the Semifinite Setting
2.5 Symmetric Operator Spaces
2.6 Examples of Symmetric Operator Spaces
2.7 Traces on Symmetric Operator Spaces
2.8 Notes
II General Theory
3 Symmetric Operator Spaces
3.1 Introduction
3.2 Submajorization in the Finite-dimensional Setting
3.3 Hardy-Littlewood(-Polya) Submajorization
3.4 Uniform Submajorization
3.5 Symmetric Operator Spaces from Symmetric Function Spaces
3.6 Symmetric Function Spaces from Symmetric Sequence Spaces
3.7 Notes
4 Symmetric Functionals
4.1 Introduction
4.2 Jordan Decomposition of Symmetric Functionals
4.3 Lattice Structure on the Set of Symmetric Functionals
4.4 Lifting of Symmetric Functionals
4.5 Figiel-Kalton Theorem
4.6 Existence of Symmetric Functionals
4.7 Existence of Fully Symmetric Functionals
4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different
4.9 Symmetric Functionals on Symmetric Operator Spaces
4.10 How Large is the Set of Symmetric Functionals?
4.11 Notes
5 Commutator Subspace
5.1 Introduction
5.2 Normal Operators in the Commutator Subspace
5.3 Normal Operators in the Closed Commutator Subspace
5.4 Subharmonic Functions on Matrix Algebras
5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace
5.6 Description of the Commutator Subspace
5.7 Commutator Subspace of the Weak Ideal
5.8 Notes
6 Dixmier Traces
6.1 Introduction
6.2 Extended Limits
6.3 Dixmier Traces on Lorentz Ideals
6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces
6.5 Dixmier Traces on Fully Symmetric Ideals of L(H)
6.6 Relatively Normal Functionals
6.7 Wodzicki Representation of Dixmier Traces
6.8 Notes
III Traces on Lorentz Ideals
7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals
7.1 Introduction
7.2 Distribution Formulas for Dixmier Traces
7.3 Lidskii Formulas for Dixmier Traces
7.4 Special Cases and Counterexamples
7.5 Diagonal Formulas for Dixmier Traces Fail
7.6 Notes
8 Heat Kernel Formulas and ζ-function Residues
8.1 Introduction
8.2 Heat Kernel Functionals
8.3 Fully Symmetric Functionals are Heat Kernel Functionals
8.4 Generalized Heat Kernel Functionals
8.5 Reduction of Generalized Heat Kernel Functionals
8.6 ζ-function Residues
8.7 Not Every Dixmier Trace is a ζ-function Residue
8.8 Notes
9 Measurability in Lorentz Ideals
9.1 Introduction
9.2 Positive Dixmier Measurable Operators in Lorentz Ideals
9.3 Positive Dixmier Measurable Operators in M1,8
9.4 C-invariant Extended Limits
9.5 Positive M-measurable Operators
9.6 Additional Invariance of Dixmier Traces
9.7 Measurable Operators in
9.8 Notes
IV Applications to Noncommutative Geometry
10 Preliminaries to the Applications
10.1 Summary of Traces on L1,∞ and M1,∞
10.2 Pseudo-differential Operators and the Noncommutative Residue
10.3 Pseudo-differential Operators on Manifolds
10.4 Notes
11 Trace Theorems
11.1 Introduction
11.2 Modulated Operators
11.3 Laplacian Modulated Operators and Extension of the Noncommutative Residue
11.4 Eigenvalues of Laplacian Modulated Operators
11.5 Trace Theorem on Rd
11.6 Trace Theorem on Closed Riemannian Manifolds
11.7 Integration of Functions
11.8 Notes
12 Residues and Integrals in Noncommutative Geometry
12.1 Introduction
12.2 The Noncommutative Residue in Noncommutative Geometry
12.3 The Integral in Noncommutative Geometry
12.4 Example of Isospectral Deformations
12.5 Example of the Noncommutative Torus
12.6 Classical Limits
12.7 Notes
A Operator Results
A.1 Matrix Results
A.2 Operator Inequalities
Bibliography
Index

Citation preview

De Gruyter Studies in Mathematics 46 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Steven Lord Fedor Sukochev Dmitriy Zanin

Singular Traces Theory and Applications

De Gruyter

Mathematical Subject Classification 2010: 46L51, 47L20, 58B34, 47B06, 47B10, 46B20, 46E30, 46B45, 47G10, 58J42.

ISBN 978-3-11-026250-6 e-ISBN 978-3-11-026255-1 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P TP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

This book is dedicated to the memory of Nigel Kalton. Nigel was going to be a coauthor but, tragically, he passed away before the text could be started. The book has become a tribute. A tribute to his influence on us, and a tribute to his influence on the area of singular traces. It would not have been written without his inspiration on techniques concerning symmetric norms and quasi-nilpotent operators. A lot of the development was very recent. Singular traces is still an evolving mathematical field. Our book concentrates very much on the functional analysis side, which we feel is heading toward some early form of maturity. It was a sense of consolidation, that we had something worth reporting, which sponsored the idea of writing a book. We thought that new results discovered with Nigel, based on previous work about symmetric spaces, symmetric functionals and commutator subspaces by Nigel, Peter and Theresa Dodds, Ken Dykema, Thierry Fack, Tadius Figiel, Victor Kaftal, Ben de Pagter, Albrecht Pietsch, Aleksander Sedaev, Evgenii Semenov, Gary Weiss, Mariusz Wodzicki, and many others, deserved a wider audience amongst operator algebraists, functional analysts, noncommutative geometers and allied mathematical physicists. Despite the patchiness of a young field, what is emerging is that singular traces are as fundamental to the study of functional analysis as the canonical trace. The book is complete, in the limited sense that any investigation can be complete, as a story of the existence of continuous traces on symmetrically normed ideals. It is a complete story about the Calkin correspondence and the Lidskii eigenvalue formula for continuous traces. It is a complete story of the Dixmier trace on the dual of the Macaev ideal. It is a complete story about Connes’ Trace Theorem. We strongly emphasize, however, that this book is not a reference for, nor a snapshot of, the entire field. We have included end notes to each chapter, giving historical background, credit of results, and reference to alternative approaches to the best of our knowledge; we apologize in advance for possible omissions. We hope we have made the topic accessible, as well as displaying what we think to be the vital and interesting features of singular traces on symmetric operator spaces. The authors acknowledge their families for their invaluable support. Steven Lord and Fedor Sukochev thank their partners, Rebekkah Sparrow and Olga Lopatko. We thank Fritz Gesztesy for the introduction to the publisher and assistance during the publication process, and for his help with the permission to reproduce a tribute from the Nigel Kalton Memorial Website. We thank Anna Tomskova for LaTeX and editing support.

vi

Preface

We thank Albrecht Pietsch for helping us with historical comments. We thank Jacques Dixmier for his permission to quote from the letter he wrote to the conference “Singular Traces and Their Applications” (Luminy, January 2012). We thank colleagues that have given us invaluable advice, encouragement, and criticism, Alan Carey, Vladimir Chilin, Victor Gayral, John Phillips, Denis Potapov, and Adam Rennie, amongst those not already mentioned, and we acknowledge contributors and co-workers in the field, Nurulla Azamov, Daniele Guido, Bruno Iochum, Tommaso Isola, Alexandr Usachev, and Joseph Várilly, amongst many others. Sydney, Australia, 30 July 2012

Steven Lord Fedor Sukochev Dmitriy Zanin

Notations

N Z ZC R RC Rd jj C MN .C/ Tr.A/ det.A/ L N

set of natural numbers set of integers set of non-negative integers field of real numbers set of positive real numbers Euclidean space of dimension d Euclidean norm on Rd field of complex numbers algebra of square N  N complex matrices trace of a matrix A determinant of a matrix A

?   C  a˛ " a (a˛ # a)

direct sum of Banach spaces, traces or operators tensor product of von Neumann algebras, traces or operators convolution orthocomplement submajorization uniform submajorization uniform convergence increasing (decreasing) net with respect to a partial order

 logC .x/

Gamma function maxflog jxj, 0g

H h, i kk k  k1 L.H / feng1 nD0 ˝ 1 diag.a/ ŒA, B A jAj

complex separable Hilbert space inner product (complex linear in the first variable) norm (usually the vector norm on a Hilbert space) operator (uniform) norm algebra of bounded operators on H orthonormal basis of H one-dimensional operator on H defined by . ˝ /x :D hx, i identity map on H diagonal operator on H associated with a 2 l1 commutator of operators A and B adjoint of the operator A absolute value of the operator A

viii

Notations

A0 AC (A ) 0 .t / and

Z



Z

t

.s, x/ds < 1g

0

1

M .0, 1/ :D fx 2 S.0, 1/ :

.t , x/d .t / < 1g. 0

Define the corresponding Lorentz operator spaces by setting M .M, / :D fA 2 S.M, / : .A/ 2 M .0, 1/g and M .M, / :D fA 2 S.M, / : .A/ 2 M  .0, 1/g. Equipped with the norms kAkM

1 :D sup t >0 .t /

Z

t

.s, A/ds, 0

66

Chapter 2 Preliminaries on Symmetric Operator Spaces

and

Z kAkM :D

1

.t , A/d .t /, 0

the Lorentz operator spaces become fully symmetric operator spaces. For the pair .L.H /, Tr/, the symmetric operator spaces M :D M .L.H /, Tr/ and M :D M .L.H /, Tr/ are symmetric ideals of compact operators, and, in fact, are the Lorentz ideals of compacat operators associated to in Example 1.2.7. Observe that the unit ball in M .M, / consists of those A 2 S.M, / with .A/  0 . So, when 0 2 L1 .RC /, then M .M, /  M. A Lorentz operator space satisfying this condition is called a Lorentz operator ideal. The reader should note that, unless .M, / is appended as in the notation M .M, /, then the notation M with no appended term will always denote the Lorentz ideal of compact operators M .L.H /, Tr/. For the studies of symmetric function (respectively, operator) spaces which generalize classes of Lp -spaces and Lorentz M -spaces we refer to [10, 139] (respectively, [68]). Not every symmetric operator space is fully symmetric. To show such a distinction it is sufficient to show that there is a symmetric function space which is not fully symmetric. Example 2.6.11. The least symmetric subspace (closed in the norm k  kM ) containing the element 0 of a Lorentz space M D M .0, 1/ with the function satisfying the condition .2t / D1 (2.5) lim t !0 .t / does not coincide with M and is, therefore, not fully symmetric. The details can be found in [139, Lemma II.5]. The preceding example is an example of a strongly symmetric function space. Symmetric function spaces that are not even strongly symmetric are mentioned in the end notes.

Normal Functionals and Trace Duality for Lorentz Operator Spaces Let .E.M, /, k  kE / be a symmetric operator space and let ' belong to the Banach dual E.M, / . Recall that ' is bounded in the dual norm k'kE  :D sup j'.A/j. kAkE 1

All the -finite projections in M belong to E.M, / since .L1 \ L1 /.M, /  E.M, /. Denote by E.M, /0 the closure of the linear span of -finite projections

Section 2.6 Examples of Symmetric Operator Spaces

67

in M in the norm k  kE . Then E.M, /0 is a symmetric operator space called the separable part of E.M, / (also termed the regular part in Simon [222]). The condition of continuity makes the following equivalent statements evident. Lemma 2.6.12. For a continuous linear functional ' 2 E.M, / the following statements are equivalent. (a) ' vanishes on the set of -finite projections in M. (b) ' vanishes on .L1 \ L1 /.M, /. (c) ' vanishes on E.M, /0 . Continuous linear functionals that satisfy one of the above criteria are called singular. Definition 2.6.13. A linear functional ' 2 E.M, / is called (a) normal if fX˛ g˛2I  E.M, / and X˛ # 0 implies that '.X˛ / ! 0. (b) completely additive if A 2 E.M, / and fp˛ g˛2I  Proj.M/ with p˛ # 0 implies that '.Ap˛ / ! 0 and '.p˛ A/ ! 0. (c)

singular if ' vanishes on E.M, /0 .

Where required we use the notation E.M, /n and E.M, /s for the sets of all normal and singular functionals from E.M, / . Evidently, if E.M, / D E.M, /0 , then E.M, / admits no nontrivial singular functional. The following theorem (established in [71]) extends the notion of trace duality, as seen earlier between L1 .M, / and the set of normal linear functionals on L1 .M, /, to Lorentz operator spaces. Theorem 2.6.14. For ' 2 M .M, / the following statements are equivalent. (a) ' is normal. (b) ' is completely additive. (c) There exists A 2 M .M, / such that '.X/ D .AX/,

8X 2 M .M, /,

and k'kM D kAkM . The reader may think about the Lorentz operator space M .M, / and its trace duality with M .M, / as a generalization of L1 .M, / and its trace duality with L1 .M, /. This result is a generalization of the Radon–Nikodym theorem, and identifies all continuous normal linear functionals on M .M, /.

68

Chapter 2 Preliminaries on Symmetric Operator Spaces

The study of non-normal continuous linear functionals on Lorentz spaces reduces to the study of singular continuous linear functionals. We cite the following special case of the famous Yosida–Hewitt decomposition for more general fully symmetric spaces (see e.g. [64, 72, 73]). Theorem 2.6.15. Every ' 2 M .M, / admits a unique decomposition ' D 'n C 's ,

'n , 's 2 M .M, /

(2.6)

where 'n is normal, in particular, there exists A 2 M .M, / such that 'n .X/ D .AX/, for all X 2 M .M, /, and 's is singular. The new aspect with Lorentz operator spaces is that continuous singular linear functionals exist on Lorentz operator spaces. Instead of the integration theory based on the normal trace  on L1 .M, /, we can replace  with a continuous singular trace ' 2 M .M, / (as defined in the next section) instead. Prompted by the noncommutative geometry of Alain Connes the couple .M .M, /, '/ replaces the couple .M, / as the basis of a new noncommutative integration theory.

2.7 Traces on Symmetric Operator Spaces We define a trace on a symmetric operator space in analogy with a trace on a two-sided ideal of compact operators. Let M be a von Neumann algebra equipped with a fixed faithful normal semifinite trace . Recall that every symmetric operator space is an operator bimodule on M. Definition 2.7.1. Let .E.M, /, k  kE / be a symmetric operator space. A unitarily invariant functional ' 2 E.M, / is called a continuous trace. That is, '.U  AU / D '.A/ for all A 2 E.M, / and unitaries U 2 M. Since a symmetric operator space E.M, / contains the set of -finite projections of the von Neumann algebra M, a positive trace (we have not requested that a continuous trace be positive in the above definition) on a symmetric operator space E.M, / is, according to the definitions in Section 2.2, a semifinite trace on M. We have already seen one important example of a continuous trace on a symmetric operator space. Example 2.7.2. The noncommutative L1 -space L1.M, / is a symmetric operator space and the extension of the faithful normal semifinite trace   : L1.M, / ! C is a continuous trace on the noncommutative L1-space. If M is a factor, then  is the unique normal trace (up to a constant) on L1 .M, /.

Section 2.7 Traces on Symmetric Operator Spaces

69

In Chapter 4, we develop the theory of symmetric functionals on symmetric operator spaces. We are especially interested in those functionals which are monotone with respect to Hardy–Littlewood(–Polya) submajorization. Definition 2.7.3. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . Let E.M, / be a symmetric operator space. The functional ' 2 E.M, / is called (a) symmetric if '.A/ D '.B/ whenever .A/ D .B/ for 0 A, B 2 E.M, /. (b) fully symmetric if '.B/ '.A/ whenever .A/  .B/ for 0 A, B 2 E.M, /. Observe that we do not request in Definition 2.7.3 (a) that ' is necessarily a positive functional on E.M, /. However, Definition 2.7.3 (b) requires that fully symmetric functionals be positive since 0  A if 0 A. The Dixmier traces introduced in equation (1.4) in Section 1.3.2, by their construction, are examples of fully symmetric singular traces. The normal trace  is a fully symmetric trace on L1 .M, /. The following lemma shows the relevance of symmetric functionals to trace theory. Lemma 2.7.4. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . Let E.M, / be a symmetric operator space. (a) Every symmetric functional on E.M, / is a continuous trace. (b) If M is atomless (or atomic) and a factor, then every continuous trace on E.M, / is a symmetric functional. Proof. The first assertion is rather trivial, but the second is not. (a) If 0 A 2 E.M, / and if U 2 M is unitary, then .U  AU / D .A/. Since ' is symmetric, it follows that '.U  AU / D '.A/. By linearity, ' is a trace. (b) If 0 A, B 2 E.M, / are such that .A/ D .B/, then there exist 0 An , Bn 2 E.M, / such that An ! A, Bn ! B and such that .An / D .Bn / are countably-valued functions. By construction, the operators An and Bn have identical discrete spectrum and their corresponding eigenprojections have the same trace. Since M is a factor, it follows that there exists a partial isometry Un which conjugates the corresponding eigenprojections of An and Bn . Consequently, An D UBn U  and, therefore, '.An / D '.Bn /. Since ' is continuous, it follows that '.A/ D '.B/. As a result, symmetric functionals express the conceptual boundary of continuous traces that may be constructed from singular values. We shall show in Chapter 4 that the construction formula always takes the form of a symmetric functional on a symmetric function space. In this way symmetric functionals express the most general

70

Chapter 2 Preliminaries on Symmetric Operator Spaces

possible scope of the Dixmier-style construction of traces. In particular, Lemma 2.7.4 says that, for the von Neumann algebra M D L.H /, every continuous trace on every symmetrically normed ideal of compact operators is a symmetric functional. That M is a factor in Lemma 2.7.4 (b) is the natural limitation on the bijective association between symmetric functionals and traces, as was the case for the Calkin correspondence between Calkin function spaces and operator bimodules. In the disintegration of a general von Neumann algebra into factors, the center (where the concept of trace is moot anyway) interrupts the bijective association, as the next example makes clear. Example 2.7.5. If M D l1 , then every continuous linear functional on l1 is a trace. However, there are no symmetric functionals on l1 . Continuous traces and symmetrical functionals can be further refined into normal and singular traces and functionals. Definition 2.7.6. Let .E.M, /, k  kE / be a symmetric operator space. A continuous trace ' 2 E.M, / is called (a) normal if ' 2 E.M, /n , that is if X˛ # 0 2 E.M, / then '.X˛ / ! 0. (b) singular if ' 2 E.M, /s , that is if ' vanishes on E.M, /0 . Normal and singular symmetric functionals are similarly defined. Since L1 .M, / D L1 .M, /0 is its own separable part, then the noncommutative L1-space admits the normal continuous trace , but does not admit any singular continuous trace. This does not represent the universal case however. Many symmetric operator spaces admit continuous singular traces. As mentioned, a study of continuous singular traces on these symmetric operator spaces leads to new possibilities for noncommutative integration theory. Next, in Part II, we consider if and only if conditions for a symmetric operator space to admit a continuous singular trace, and the form of those traces. We will show that continuous traces decompose uniquely into normal and singular parts, providing a trace version, for all symmetric operator spaces, of the Yosida–Hewett decomposition. We study fully symmetric traces on Lorentz operator ideals in detail in Part III. Unless stated explicitly, every trace we consider on a symmetric operator space is assumed to be continuous.

2.8 Notes Von Neumann Algebras and Semifinite Normal Traces The uniform, strong and weak operator topologies, and rings of bounded operators closed in those topologies, were introduced by F. Murray and J. von Neumann [166]. Along with C  algebras, von Neumann algebras form one of the pillars of operator algebra theory [63, 243].

Section 2.8 Notes

71

The cited monographs, amongst many others, deal comprehensively with the theory of von Neumann algebras and semifinite normal traces. Theorem 2.1.3 is proved in [217]. Singular Values in Semifinite von Neumann Algebras Singular values for semifinite von Neumann algebras can be traced to the fundamental paper by F. Murray and J. von Neumann [166] and to A. Grothendieck’s announcement [102] (in the special case of finite atomless factors). The general case was almost simultaneously considered by M. Sonis [223] and V. I. Ovchinnikov [174,175]. Later it was studied by F. J. Yeadon [263], T. Fack [88], and T. Fack and H. Kosaki in [90]. A presentation of the subject is given in the forthcoming book [68]. Another frequently used notation for the set S.M, / of -measurable operators is f M. We refer the reader to [168] for the proof of Lemma 2.3.4 and Theorem 2.3.5. The algebra S.M, /, when equipped with the (so-called) measure topology, is a complete topological algebra and it forms the basis of the theory of noncommutative integration, playing the same role that the algebra S.0, 1/, equipped with the Lebesgue measure, plays in classical integration theory. See [68, 90, 168, 174, 175]. The equation (2.2) is a noncommutative analog of the classical formula linking the distribution function of a random variable and its decreasing rearrangement [10,139]. The proof of the noncommutative version can be found in [174, Theorem 1], [88, Proposition 1.3], [90, Proposition 2.2] and [68]. The formula given in Example 2.3.9 is classical and still holds for rearrangements of measurable functions defined on an arbitrary -finite measure space [10, 68, 139]. The proof of Lemma 2.3.12 can be found in [68, 88, 90, 174, 263]. The proof of Theorem 2.3.13 may be found in [174, Theorem 1], [90, Proposition 2.4]. The result of Lemma 2.3.15 is well known (see [3] and [39]) and is frequently used. Calkin Correspondence in the Semifinite Setting The Calkin correspondence for general factors is folklore. D. Guido and T. Isola in [103, Section 3] provide a similar statement. Symmetric Operator Spaces Question 2.5.4 naturally arises from J. von Neumann’s paper [172] published in 1937. In this paper von Neumann introduced symmetric norms in the finite-dimensional setting, in particular, for the matrix ideals Lp , five years after Banach’s fundamental book [8] introduced and studied in depth the lp -sequence and Lp -function spaces. In contrast to Banach’s book, the von Neumann paper [172] is almost completely unknown, even to experts. It appeared in an obscure Russian journal, which ceased to exist almost immediately after its first volume was published (the coming Second World War and disruption of scientific contacts did not help either). However, from the present point of view the theory of symmetric operator spaces began with this paper. In fact, von Neumann anticipated the Calkin correspondence by associating with every n-dimensional symmetric sequence space a n2 -dimensional symmetric matrix space (coinciding as a linear space with the set of all n  n matrices). This connection inspired the theory of symmetrically normed ideals of compact operators, developed by von Neumann with R. Schatten [205–208], and later by Gohberg and Krein [98, 99]. The term “symmetrically normed ideal of compact operators” is due to the highly influential monographs [98, 99]. B. Simon in his recommended book [222] prefers a somewhat vaguer term “trace ideals”. Sometimes, there is a difference in the terminology. For example,

72

Chapter 2 Preliminaries on Symmetric Operator Spaces

in Theorem 1.16 of [222] the assertion .b/ does not hold for the norm of an arbitrary symmetrically normed ideal (see e.g. corresponding counterexamples in [129, p. 83]). The Question 2.5.4, which is the infinite-dimensional version of von Neumann’s initial matrix result, was fully answered in [129]. An improved version of this result is given in Chapter 3. For a thorough survey describing the modern state of this theory we refer to [185]. The general theory of symmetric operator spaces is studied in numerous papers, among which we cite just a few [41, 69–71, 230–234, 263, 264] as the most relevant to our present exposition. For proofs of the Banach space structure of the noncommutative Lp -spaces we refer to [61, 105, 140, 168, 217]. The -compact operators in the semifinite setting was considered by M. Sonis [223] and V. I. Ovchinnikov [176]. A different notion of compact operators in the semifinite setting originated with M. Breuer [21, 22]. For a beautiful exposition of the theory of symmetric function spaces and its application in interpolation theory, we refer to [139] and [10]. These books contain a concise exposition of the theory of the Lorentz function spaces M  and M , as well as rich historical information and references on earlier works in this area. In particular, on the contribution of G. G. Lorentz [156], which led to the introduction of these spaces. These works undoubtedly influenced researchers studying noncommutative analogs of these spaces. We observe that in many sources the spaces M are called Marcinkiewicz spaces (see e.g. [139] and many of our own papers [33–35, 153, 241]). In this book, however, we follow the terminology of [10] and refer to those spaces as Lorentz spaces. The first example of a symmetric operator ideal which is not fully symmetric is due to G. Russu [200]. It is much harder to find a symmetric operator ideal which is not strongly symmetric. In the setting of symmetric function spaces on .0, 1/ such an example is known and due to A. Sedaev [211], who answered a commutative variant of this question (asked by one of the authors in 1985). Sedaev’s construction is technical and heavily depends on some remarkable facts from the theory of Lorentz function spaces contained in the paper [20] of M. Sh. Braverman and A. A. Mekler. A noncommutative version of Sedaev’s result yielding an example of a symmetric operator space on a II1 -factor which does not admit a strongly symmetric norm was presented in [233]. Fully symmetric ideals of compact operators are Banach ideals of compact operators that are solid under Hardy–Littlewood(–Polya) submajorization . Two-sided ideals of compact operators that are solid under the submajorization  (with no Banach space structure assumed) have been studied under the term arithmetically mean closed ideals, see [80, 118, 121]. Trace Duality, Normal and Singular Functionals In the framework of the theory of Banach lattices, it is customary to consider a decomposition of an arbitrary continuous functional on a given Banach lattice into a direct sum of “normal” (that is continuous with respect to order convergence) and “singular” parts. This decomposition is usually linked with the classical theorem of K. Yosida and E. Hewitt [265], which states that any bounded additive measure can be uniquely represented as the sum of a countably additive measure and a purely finitely additive measure, the so-called singular part, which is characterized by the fact that its absolute value does not dominate any non-zero positive countably additive measure. It is interesting to observe that this purely commutative result has a noncommutative precursor. Indeed, a noncommutative analog of this result was obtained by J. Dixmier [60]. Dixmier mentions that some of his results had been obtained ear-

73

Section 2.8 Notes

lier in papers of R. Schatten and/or R. Schatten and J. von Neumann [205, 207, 208], where it was shown that, using trace duality, the Banach dual of the space of compact operators in a Hilbert space is the trace class operators, and the Banach dual of the trace class operators is the space of all bounded operators. Dixmier’s result, which is based on this duality, states that each bounded linear functional f on the algebra L.H / of all bounded linear operators on a separable Hilbert space is a direct sum of a trace functional g and a singular functional h, vanishing on the compact operators, such that kf k D kgk Ckhk. A direct analog of this result for a general von Neumann algebra M was first established by M. Takesaki [242]. Takesaki’s result yields that for anyL W  -algebra M the following decomposition of the conjugate space   M holds: M D M M?  , where the direct sum is taken in the l1 -sense. In the framework of the approach above, it is natural to study further the singular component of the decomposition. In the Banach lattices setting, the papers by G. Ya. Lozanovskii [157– 159] studied the “singular” part .M  /s of the dual to the commutative analogs M of Lorentz operator ideals. This work did not directly lead to the notion of “singular traces” but was nevertheless helpful to recognize the connections between commutative and noncommutative theories (see [67]). For symmetrically normed ideals of compact operators a complete analogy to the Yosida and Hewitt result was achieved in the paper [73]. Proposition 2.7 in [73] extended Dixmier’s earlier results on L.H /. For extensions to (fully and strongly) symmetric operator spaces on arbitrary von Neumann algebras we refer the reader to [72] and [64]. It is interesting to observe that the assertion of Theorem 2.6.15 falls short of the l1 -type decomposition achieved in [242,265] and [73]. It follows from the proof of [64, Proposition 5.5] (see also comments following Theorem 5.9 in that paper) that we have for the decomposition (2.6) k'n kM , k's kM 4k'kM .

Part II

General Theory

This part of the book discusses the general theory of symmetric functionals on symmetric operators spaces. Chapter 3 confirms that the Calkin correspondence, for atomless or atomic semifinite von Neumann algebras, is a bijective functor between symmetric operator spaces and symmetric function spaces. The technique used is a recent development, and solves a question about symmetrically normed ideals of compact operators which goes back to works of J. von Neumann and R. Schatten. That the Calkin correspondence is a functor of Banach spaces is the necessary prerequisite for discussing the correspondence between symmetric functionals on symmetric operator spaces (which we recall are continuous linear functionals) and symmetric functionals on symmetric function spaces. Chapter 4 confirms that the association between symmetric functionals on operator spaces and symmetric functionals on function spaces is given by the Calkin correspondence. In summary, the situation shown in Chapter 3 and Chapter 4 is the following. (a) If .E, k  kE / is a symmetric function space then the assignment E.M, / :D fA 2 S.M, / : .A/ 2 Eg with the norm kAkE :D k.A/kE is a symmetric operator space for a von Neumann algebra M equipped with a fixed faithful normal semifinite trace . Moreover, every symmetric functional f 2 E  lifts to a symmetric functional ' 2 E.M, / according to the formula '.A/ D f ..A//,

A  0.

(b) Conversely, if M is atomless (or atomic) and .E.M, /, k  kE / is a symmetric operator space then the assignment E :D fx 2 S : .x/ D .A/ for some A 2 E.M, /g (here S is the function space in Example 2.3.3 of Chapter 2) with the norm kxkE :D kAkE

76

Part II

General Theory

is a symmetric function (or sequence) space. Moreover, for every symmetric functional ' 2 E.M, / there exists a symmetric functional f 2 E  such that '.A/ D f ..A//,

A  0.

Chapter 4 continues by addressing the implications of this result for continuous traces on symmetric operator spaces (recall that every symmetric functional is a continuous trace on E.M, /). Specifically, existence of symmetric functionals, existence of fully symmetric functionals, existence of symmetric functionals that are not fully symmetric, and the question of whether the functionals are normal or singular, unique or a plethora, are answered from a study of symmetric functionals on function spaces. The main results are Theorem 4.9.2 and Theorem 4.10.1. As an example of the use of Theorem 4.9.2 and Theorem 4.10.1 we can answer the basic existence questions of continuous traces on Banach operator bimodules of semifinite atomless (or atomic) factors (recall from Lemma 2.4.6 that the symmetric operator spaces and the Banach bimodules of a factor are in bijective correspondence, and from Lemma 2.7.4 that the set of symmetric functionals on a symmetric operator space of a factor is identical to the set of all continuous traces). If M is an atomless (or atomic) semifinite factor and E.M, / is a symmetric operator space equipped with a Fatou norm, then one of the following mutually exclusive possibilities hold. (a) The space E.M, / does not admit a nontrivial continuous trace. (b) The space E.M, / admits a (up to a constant factor) unique nontrivial continuous trace and it is the (extension of the) faithful normal semifinite trace . (c) The space E.M, / admits an infinite number of nontrivial continuous singular traces (in fact, the set of Hermitian continuous traces is an infinite dimensional Banach lattice). As an example of (b), see the extension of  to the noncommutative L1-space L1.M, / in Example 2.6.2. Indeed, if E.M, / 6 L1 .M, / is not a subset of the noncommutative L1 -space and  is infinite then (b) is impossible, and the case (a) or (c) can be determined by the result that (a) does not hold if there exists a positive operator A 2 E.M, / such that limm!1 m1 kA˚m kE > 0. For the factor M D L.H /, in existing terminology, Chapter 3 and Chapter 4 establish that (a) there is no ambiguity in using the notation for the Calkin correspondence used in Chapter 1 (as a functor between sets) for the Calkin correspondence  : EC ! EC as a functor between Banach spaces.

Part II

77

General Theory

(b) any continuous trace ' on a symmetric ideal of compact operators E is obtained from a symmetric functional f on the corresponding Calkin sequence space E according to the formula '.A/ D f ı .A/,

A  0.

(II.1)

Formula (II.1) reduces the study of continuous traces on symmetric ideals of compact operators to the study of symmetric functionals on sequence spaces. It raises two further questions for the theory of continuous traces on symmetric ideals of compact operators, which are addressed in Chapter 5 and Chapter 6. In Chapter 5 we show that the Lidskii formula holds for all continuous traces. If E is a symmetric ideal of compact operators and E is the corresponding Calkin sequence space, then an eigenvalue sequence .A/ 2 E for any A 2 E, or  : E ! E, and the Lidskii formula holds for every continuous trace ' on E, '.A/ D f ı .A/. In this case the correspondence between continuous traces and symmetric functionals is unequivocal: ' D f ı  , f D ' ı diag. Here diag is the diagonal operator for any orthonormal basis of H from Chapter 1. These result are obtained by the commutator subspace method of Kalton, Kalton and Dykema, and Dykema, Figiel, Weiss and Wodzicki. We also introduce the closed commutator subspace, which is an essential concept for the study of continuous traces (we discuss this in more detail in Chapter 5 and its notes). The second question raised by (II.1) is exactly what do symmetric functionals on symmetric sequence spaces look like? It is nice to have an existence result, but what concrete formulas of singular values give rise to continuous traces? We can answer this question for fully symmetric functionals. Dixmier traces, introduced in Section 1.3.2 of Chapter 1, are an example of a concrete construction. Chapter 6 looks initially at Lorentz operator ideals of L.H /, which fall into the category of fully symmetric operator spaces with Fatou norms, and introduces Dixmier’s construction on a general Lorentz ideal of compact operators. For a Lorentz ideal M , the criteria on existence of symmetric functionals translate to the condition .2t / lim inf D 1. t !1 .t / If the condition is satisfied, the Lorentz ideal possesses an infinite number of continuous traces, otherwise none. The first interesting result is that the set of normalized

78

Part II

General Theory

fully symmetric functionals on these Lorentz ideals and the set of Dixmier traces on them are identical. The fully symmetric functionals are therefore fully constructible. Even more surprisingly, we show that Dixmier traces can be defined on an arbitrary fully symmetric ideal of the algebra L.H /, and that any fully symmetric functional on a fully symmetric ideal is approximated in the weak topology by Dixmier traces. Thus, all fully symmetric functionals are asymptotically constructible according to, basically, Dixmier’s original construction. The construction in Chapter 6 extends some trace results of Wodzicki, but we discuss that in more detail in that chapter and its notes.

Chapter 3

Symmetric Operator Spaces

3.1 Introduction In this chapter we introduce uniform submajorization, which extends the classical notion of Hardy–Littlewood(–Polya) submajorization used extensively in the theory of symmetric function spaces. Uniform submajorization proves an indispensable tool for showing the main results of this chapter and the next, and is one of the central technical devices in symmetric operator space theory. Specifically, the main result of this chapter is the affirmative answer to the Questions 2.5.4 and 2.5.5 posed in Section 2.5 of Chapter 2. We know already from Theorem 2.5.3, that if E.M, / is a symmetric operator space for an atomless (or atomic) semifinite von Neumann algebra M then the assignment E :D fx 2 S : .x/ D .A/ for some A 2 E.M, /g and kxkE :D kAkE for all x 2 E, A 2 E.M, / such that .A/ D .x/ defines a symmetric function (or sequence) space .E, k  kE /. This is the relatively easy direction for the Calkin correspondence as a functor between symmetric operator spaces and symmetric function spaces. The converse, Question 2.5.4, represents the more difficult direction. The theorem below was recently proved for M D L.H / in [129] using uniform submajorization, and we provide a systematic, and extended, account of the proof in the following sections. Theorem 3.1.1. Let M be an atomless (or atomic) von Neumann algebra equipped with a faithful normal semifinite trace . (a) If M is atomless and if .1/ D 1, then E :D fx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g , kxkE :D kAkE and E.M, / :D fA 2 S.M, / : .A/ 2 Eg , kAkE :D k.A/kE is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric function space on .0, 1/ and a symmetric operator space of .M, /.

80

Chapter 3 Symmetric Operator Spaces

(b) If M is atomless and if .1/ D 1, then E :D fx 2 S.0, 1/ : .x/ D .A/ for some A 2 E.M, /g , kxkE :D kAkE and E.M, / :D fA 2 S.M, / : .A/ 2 Eg , kAkE :D k.A/kE is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric function space on .0, 1/ and a symmetric operator space of .M, /. (c) If M is atomic, then E :D fx 2 l1 : .x/ D .A/ for some A 2 E.M, /g , kxkE :D kAkE and E.M, / :D fA 2 S.M, / : .A/ 2 Eg , kAkE :D k.A/kE is a bijective correspondence .E, kkE / $ .E.M, /, kkE / between a symmetric sequence space and a symmetric operator space of .M, /. Below M is an arbitrary von Neumann algebra (not necessarily atomless or atomic) equipped with a faithful normal semifinite trace . Question 2.5.5 is answered by: Theorem 3.1.2. For every symmetric function space E on the semi-axis .0, 1/, the set E.M, / :D fA 2 S.M, / : .A/ 2 Eg equipped with the norm kAkE :D k.A/kE is a symmetric operator space of .M, /. The proofs of the theorems stated above are given in Section 3.5 and Section 3.6. In Section 3.6 a variant of Theorem 3.1.2 is given for symmetric sequence spaces.

3.2 Submajorization in the Finite-dimensional Setting Uniform submajorization differs from its Hardy–Littlewood counterpart only in the infinite dimensional setting. In this section we demonstrate the key ideas in the setting of finitely supported sequences. If x 2 Rn , then (the range of) the singular value function .x/ introduced in Section 2.3 is a decreasing rearrangement of the vector jxj. Fix x 2 RnC and define the

Section 3.2 Submajorization in the Finite-dimensional Setting

81

set C .x/ :D fy 2 RnC : y  xg of all positive sequences in Rn submajorized by x. The following lemma is well known, but we provide a short proof. Lemma 3.2.1. For every x 2 RnC , the set C .x/ is a convex polyhedron. Proof. The convexity of C .x/ follows from Lemma 3.3.3. Let Sn be the set of all permutations of the set f0, 1, : : : , n  1g, For every permutation  2 Sn , consider the convex polyhedrons A :D fy 2 RnC : y..0//  y..1//      y..n  1//g, which is an intersection of 2n  1 half-planes, and B :D fy 2

RnC

:

k X

y..l//

lD0

k X

.l, x/, 0 k n  1g,

lD0

which is an intersection of 2n half-planes. We have [  \ A B . C .x/ D  2Sn

Every set A \ B ,  2 Sn , is a convex polyhedron. Hence, C .x/ is a union of at most nŠ convex polyhedrons. However, C .x/ is convex and, therefore, connected. It follows that C .x/ is a polyhedron. Every polyhedron has a finite number of extreme points. The following theorem describing the extreme points of the polyhedron C .x/ is folklore. Theorem 3.2.2. The only extreme points of C .x/ are those y 2 Rn with .y/ D .x/ Œ0,m for some m < n. Proof. We prove the assertion using induction on n. Let y 2 C .x/ be an extreme point. Without loss of generality, y D .y/ and x D .x/. Setting m to be the largest value such that .m, y/ > 0, we have y 2 C .x Œ0,m /. Thus, y is an extreme point of the set C .x Œ0,m /. If m < n  1, then the assertion follows by induction. From now on, we assume that m D n  1. It follows that .l, y/ > 0 for 0 l < m. Suppose first that min 0k 0 . ˛ :D inf k Since k  0, 0 k m, it follows that k  ˛k  0 for 0 k m and xD

m X

.k  ˛k /xk ,

kD0

m X

.k  ˛k / D 1.

kD0

At least one of the coefficients in the above sum is 0. Rearranging the summands, if necessary, we obtain xD

m1 X kD1

0k xk ,

m1 X

0k D 1

kD0

with positive numbers 0k , 0 k m  1. Repeat the argument until m does not exceed n. We are ready for the key result of this section, which describes the set of positive sequences in Rn submajorized by a fixed positive sequence x 2 Rn . Corollary 3.2.4. If x 2 RnC , then every y 2 C .x/ is a convex combination of at most .n C 1/ sequences dominated by x. That is, yD

n X

k zk ,

.zk / .x/, 0 k n.

kD0

Proof. The assertion follows from Theorem 3.2.3 and Theorem 3.2.2.

3.3 Hardy–Littlewood(–Polya) Submajorization We introduce submajorization in the sense of Hardy, Littlewood and Polya and we refer an interested reader to their classic book [109]. Another much more modern and very influential book by A. W. Marshall and I. Olkin [161] describes the fundamental value of the submajorization theory and its influence which permeates almost every branch of modern mathematics. We especially emphasize the importance of this theory in the interpolation theory of linear operators [10, 139] and, our interest, the related theory of symmetric function spaces. We repeat Definition 2.5.6.

84

Chapter 3 Symmetric Operator Spaces

Definition 3.3.1. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1 /.M, /. The operator B is said to be submajorized by A and written B  A if Z t Z t .s, B/ds .s, A/ds, t  0. 0

0

The connection of this definition with classical submajorization theory (that is, for functions or sequences) is immediate: we have B  A , .B/  .A/. The function

Z

t

t!

A 2 S.M, /,

.s, A/ds,

t >0

0

is paramount to our study of symmetric spaces and symmetric functionals. The following lemma is an important identification. Lemma 3.3.2. Let M be an atomless semifinite von Neumann algebra. For every t 2 RC and every A 2 .L1 C L1 /.M, /, we have Z t .s, A/ds D supf.pjAjp/ : p 2 Proj.M/, .p/ t g. 0

The same assertion is valid for atomic algebras provided that t 2 ZC . Proof. It follows from Lemma 2.6.3 that Z 1 .pjAjp/ D .s, pjAjp/ds. 0

Note that we have directly from the definitions .s, pjAjp/ D 0,

8s > .p/ D t

and .s, pjAjp/ .s, jAj/, Therefore,

Z

t

.pjAjp/

8s .p/ D t .

.s, jAj/ds.

0

Now, we prove the converse inequality. Fix  > 0 and select s  0 such that njAj .s/ t njAj ..1  /s/. Select an arbitrary projection p 2 M such that EjAj .s, 1/ p EjAj ..1  /s, 1/,

.p/ D t .

85

Section 3.3 Hardy–Littlewood(–Polya) Submajorization

Observe that

Z

t

.pjAjp/  .1  /

.s, A/ds. 0

Since  > 0 is arbitrarily small, the assertion follows. The following two results are fundamental to submajorization theory. Theorem 3.3.3. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1 /.M, /. We have .A C B/  .A/ C .B/. Proof. By Lemma 2.3.15, there exist partial isometries U , V such that jA C Bj U  jAjU C V  jBjV . It is, therefore, sufficient to prove the assertion for the case when both A and B are positive. By Lemma 2.3.18, we may assume without loss of generality that M is atomless. Fix t > 0 and  > 0. By Lemma 3.3.2, there exists a projection p 2 M such that .p/ t and such that Z t .s, A C B/ds .p.A C B/p/ C . 0

By Lemma 3.3.2, we also have Z t .pAp/ .s, A/ds,

Z

.s, B/ds.

0

Thus,

Z

t 0

t

.pBp/ 0

Z

t

.s, A C B/ds

Z

t

.s, A/ds C

0

.s, B/ds C .

0

Since  is arbitrarily small, the assertion follows. The next result is a companion to Theorem 3.3.3. For s > 0, the dilation operator s acting on the space L0.0, 1/ of all measurable functions is defined by setting   t , x 2 L0 .0, 1/, t > 0. .s x/.t / :D x s Theorem 3.3.4. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1 /.M, / be positive operators. We have .A/ C .B/  21=2 .A C B/.

86

Chapter 3 Symmetric Operator Spaces

Proof. By Lemma 2.3.18, we may assume without loss of generality that M is atomless. Fix t > 0 and  > 0. By Lemma 3.3.2, there exist projections p, q 2 M such that .p/, .q/ t and Z t Z t .s, A/ds .pAp/ C , .s, B/ds .qBq/ C . 0

0

Define a projection r D p _ q. It follows that .r/ .p/ C .q/ D 2t . Hence, Z 2t .pAp/ C .qBq/ .r.A C B/r / .s, A C B/ds. 0

Therefore, Z

t

Z

t

.s, A/ds C

0

Z

2t

.s, B/ds

0

.s, A C B/ds C 2.

0

Since  > 0 is arbitrarily small, the assertion follows. We also need a slight generalization of the preceding assertion. This result provides a fundamental estimate, which will be connected to uniform submajorization in subsequent sections. Lemma 3.3.5. Let M be a semifinite von Neumann algebra. Let Ak 2 .L1 C L P11/.M, /, k 2 N, be positive operators and let ˛k 2 RC , k 2 N, be such that kD1 ˛k 1. We have  Z a  X 1 Z ˛k a 1 X .s, Ak /ds  s, Ak ds, 8a > 0. kD1

0

0

Here, we assume that the series

P1

kD1 Ak

kD1

converges in .L1 C L1 /.M, /.

Proof. By Lemma 2.3.18, we may assume without loss of generality that M is atomless. Fix  > 0. For every k 2 N, we can find a projection pk such that .pk / D ˛k a and Z ˛k a

0

Set p D

p . _1 kD1 k

1 Z X kD1

˛k a 0

Since

.s, Ak / 2k  C .pk Ak pk /.

P1

kD1 ˛k

.s, Ak /  C

1 X

1, it follows that .p/ a. It follows that

.pk Ak pk /  C

kD1

1 X

.pAk p/

kD1

   X Z 1 Ak p  C DC p kD1

a

  X 1  s, Ak ds.

0

Since  > 0 is arbitrarily small, the assertion follows immediately.

kD1

87

Section 3.3 Hardy–Littlewood(–Polya) Submajorization

The following characterization of Hardy–Littlewood submajorization is known for the classical setting M D L.H / (see [98, Chapter II, Lemma 3.4]). We present a short proof of the general result. Theorem 3.3.6. If 0 A, B 2 .L1 C L1 /.M, /, then B  A if and only if ..B  t /C / ..A  t /C /,

8 t > 0.

(3.2)

Proof. Fix t > 0. We have ..A  t /C / D ..A/  t /C. Applying Lemma 2.6.3 to the operator .A  t /C , we obtain Z 1 Z nA .t / ..A  t /C/ D ..s, A/  t /Cds D ..s, A/  t /ds. (3.3) 0

0

Computing the derivative of the function Z u u! ..s, A/  t /ds 0

we see that this function attains its maximum at u D nA .t /. If B  A, then Z nB .t / Z nA .t / Z nB .t / ..s, B/  t /ds ..s, A/  t /ds ..s, A/  t /ds. 0

0

0

Inequality (3.2) now follows from (3.3). Suppose now that (3.2) holds. Fix u > 0 and set t D .u, A/. It follows that Z nB .t / Z u ..s, B/  t /ds ..s, B/  t /ds D ..B  t /C/ 0 0 Z u ..A  t /C / D ..s, A/  t /ds. 0

Hence,

Z

u

Z .s, B/ds

0

u

.s, A/ds. 0

Since u is arbitrary, we have B  A. We complete this section with a lemma, which provides a useful submajorization estimate for the direct sum introduced in Definition 2.4.3. Lemma 3.3.7. Let M be a semifinite von Neumann algebra and let Ak 2 .L1 C L1 /.M, /, k  0, be positive operators. It follows that   Z t  M Z t  X 1 1  s, Ak ds   s, Ak ds. 0

kD0

0

kD0

88

Chapter 3 Symmetric Operator Spaces

Proof. Let M0 be the commutative von Neumann algebra constructed in Theorem 2.3.11. Choose positive operators Bk 2 S.M0 , /, k  0, such that .Bk / D .Ak /, Bk  .1, Bk /EBk .0, 1/,

k  0,

and such that Bk Bl D 0 for k ¤ l. For a given t > 0, there exists a projection p 2 M0 such that .p/ D t and   X   Z t  X 1 1  s, Bk ds D  p Bk p . 0

kD0

kD0

Define pairwise orthogonalPprojections pk :D p ^ EBk .0, 1/. Since M0 is commutative, it follows that p D 1 kD0 pk . Observe that pk Bl D pl Bk D 0 for k ¤ l and, therefore,   X  Z t  X 1 1  1 Z .pk /  X  s, Bk ds D  pk Bk pk  s, Ak ds. 0

kD0

kD0

kD0

It now follows from Corollary 3.3.5 that  Z Z t  X 1  s, Bk ds 0

Since

P

k0 .pk /

kD0

P k0

0

.pk /

0

  X 1  s, Ak ds. kD0

D t , the assertion follows.

The next section discusses uniform submajorization, which is an extension to the Hardy–Littlewood submajorization theory.

3.4 Uniform Submajorization The following definition, introduced originally in [129], plays a major role in the Calkin correspondence and in our treatment of singular traces. Definition 3.4.1. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1 /.M, /. We say that B is uniformly submajorized by A (written B C A) if there exists  2 N such that Z b Z b .s, B/ds .s, A/ds, a b. (3.4) a

a

Uniform submajorization is a stronger condition than the Hardy–Littlewood submajorization introduced in the last section. Our main objective in this chapter is Theorem 3.4.2 describing (in the cases M D l1 , M D L1 .0, 1/ and M D L1 .0, 1/) the convex hull of the set f0 C : .C / .A/g in terms of uniform submajorization.

89

Section 3.4 Uniform Submajorization

Theorem 3.4.2. Let 0 x, y 2 l1 . (a) If y belongs to the convex hull of the set f0 z : .z/ .x/g, then y C x. (b) If y C x, then, for every  > 0, the element .1  /y belongs to the convex hull of the set f0 z : .z/ .x/g. The same assertion holds for functions x, y 2 .L1 C L1 /.0, 1/ (or x, y 2 L1.0, 1/). The following corollary identifies symmetric sequence (or function) spaces as monotone with respect to uniform submajorization. This identification is our main tool in linking commutative and noncommutative symmetric spaces. Corollary 3.4.3. Let E be a symmetric sequence space and let x 2 E. If y 2 l1 is such that y C x, then y 2 E and kykE kxkE . The same assertion holds for symmetric function spaces. Proof. Fix  > 0. By Theorem 3.4.2 (b), there exist n 2 N, 0 zk 2 E, 1 k n, and positive numbers k , 1 k n, such that .zk / .x/ for every 1 k n and n n X X .1  /y D k zk , k D 1. kD1

kD1

Therefore, .1  /kykE

n X

k kzk kE

kD1

n X

k kxkE D kxkE .

kD1

Since  > 0 is arbitrarily small, the assertion follows.

Proof of Theorem 3.4.2 We start the proof with a simple lemma which strengthens Theorem 3.3.3 and Theorem 3.3.4 by replacing Hardy–Littlewood submajorization with its uniform counterpart. Lemma 3.4.4. Let M be a semifinite von Neumann algebra and let A, B 2 .L1 C L1 /.M, / be positive operators. We have .A C B/ C .A/ C .B/ C 21=2 .A C B/. Proof. It follows from Theorem 3.3.3 that Z b Z b .s, A C B/ds ..s, A/ C .s, B//ds. 0

0

90

Chapter 3 Symmetric Operator Spaces

It follows from Theorem 3.3.4 that Z 2a Z a .s, A C B/ds  ..s, A/ C .s, B//ds. 0

0

Subtracting the inequalities, we obtain Z b Z b .s, A C B/ds ..s, A/ C .s, B//ds.

(3.5)

a

2a

This proves the first inequality. The proof of the second inequality is identical. For all positive x, y 2 l1 , we set   N X xj , .xj / .x/, xj  0 . ŒŒy, x :D inf N : y j D1

We employ the same notation for positive functions x, y 2 .L1 C L1 /.0, 1/ (or x, y 2 L1.0, 1/). We study the properties of ŒŒ,  as the proof of Theorem 3.4.2 critically depends on them. The following lemma establishes the convexity of the operation y ! ŒŒy, x for an arbitrary 0 x 2 l1 . It also provides a formula for the Minkowski functional of the set Q.x/ :D the convex hull of the set of sequences f0 z : .z/ .x/g. We denote the latter functional by hz; xi :D inff > 0 : z 2 Q.x/g (and hz; xi :D 1 if z is not in the linear span of Q.x/). Lemma 3.4.5. The mapping .x, y/ ! ŒŒx, y satisfies the following properties. (a) If y1 y2 , then ŒŒy1 , x ŒŒy2 , x. If x1 x2 , then ŒŒy, x2  ŒŒy, x1 . (b) For all positive x, y1 , y2 2 l1 , we have ŒŒy1 C y2 , x ŒŒy1 , x C ŒŒy2 , x. (c) For all positive x, y 2 l1 , there exists a limit lim

N !1

1 ŒŒNy, x. N

91

Section 3.4 Uniform Submajorization

(d) For a fixed positive x 2 l1 , and every positive y 2 l1 , we have 1 ŒŒNy, x. N !1 N

hy; xi D lim

(3.6)

The same assertion holds for functions x, y 2 .L1 C L1 /.0, 1/ (or x, y 2 L1.0, 1/). Proof. The first and second properties follow directly from the definition. The third property follows from the second one and Lemma 3.4.6, which is a standard Fekete Lemma. In order to prove the fourth property, let x D .x/ and for an arbitrary  > 0, let N 2 N, cj  0, 1 j N , and xj 2 l1 , 1 j N , be such that .xj / D x for 1 j N and y

N X

cj xj ,

j D1

N X

cj < hy; xi C .

j D1

Then, for every positive integer M , we have My

N X

.ŒMcj  C 1/xj

j D1

where Œa is the integral part of a. Consequently, ŒŒMy, x

N X

.ŒMcj  C 1/ < M.hy; xi C / C N .

j D1

Conversely, for any given M 2 N, let K D ŒŒMy, x and let h1 , : : : , hK  0 be such that .hj / D .x/ D x, j D 1, 2, ..., K and My

K X

hj .

j D1

Then we have K X K 1 y hj  Q.x/, M M j D1

and, consequently, hy; xi

1 M ŒŒMy, x.

hy; xi

Hence,

1 N ŒŒMy, x hy; xi C  C . M M

92

Chapter 3 Symmetric Operator Spaces

Letting M ! 1 and keeping in mind that N is fixed and  > 0 is arbitrary, we obtain the equality (3.6). Lemma 3.4.6 (Fekete Lemma). If ak , k  1, is a positive sequence such that akCl ak C al for k, l  1, then an an D inf . lim n!1 n n0 n Proof. Fix n 2 N. We have amn man for m 2 N. Thus, for k D mnCs, 0 s < n, we have amn C as man C as ak . k mn C s mn If k ! 1, then m ! 1 and, therefore, ak an . k n

lim sup k!1

Taking the infimum over n 2 N, we obtain the assertion. The following lemma shows that the operation ŒŒ,  is compatible with the direct sum operation. Lemma 3.4.7. If xn , yn 2 l1 , n  1, are positive sequences, then

 M 1 1 M yn , xn sup ŒŒyn , xn . nD1

n2N

nD1

The same assertion holds for positive functions xn , yn 2 .L1 CL1 /.0, 1/ (or xn , yn 2 L1.0, 1/), n  1. Proof. Set Nn D ŒŒyn , xn ,

N D sup ŒŒyn , xn  D sup Nn . n2N

n2N

If N D 1, then the assertion is trivial. Suppose that N < 1. For every n 2 N, there exist xnk 2 l1 , 1 k Nn , such that .xnk / .xn / and yn

Nn X

xnk .

kD1

Setting xnk D xn for Nn < k N , we have 1 M nD1

yn

1 X N M nD1

kD1

1 N M X xnk D xnk . kD1 nD1

93

Section 3.4 Uniform Submajorization

Since 

1 M

1 M xnk  xn ,

nD1

nD1

the assertion follows. The following lemma is a finite-dimensional result based on Corollary 3.2.4. Lemma 3.4.8. Let x, y 2 RN C be such that y  x. For every M 2 N, we have ŒŒMy, x M C N C 1. Proof. Since y 2 C .x/, it follows from Corollary 3.2.4 that there exists a sequence xk , 0 k N , of extreme points of C .x/ and a sequence of positive numbers k , 0 k N , such that N X

yD

 k xk ,

1D

kD0

N X

k .

kD0

By Theorem 3.2.2, .xk / .x/ for every 0 k N . Hence, there exist zk  xk such that .zk / D .x/. It clearly follows that My D

N X

M  k xk

kD0

N X

.1 C ŒM k /zk .

kD0

Therefore, ŒŒMy, x

N X

.1 C ŒM k / M C N C 1.

kD0

The following lemma is an infinite-dimensional surrogate for the preceding result. Lemma 3.4.9. Let x D .x/ 2 l1 and y D .y/ 2 l1 be such that y  x Œu,1/ for some u 2 N. If jsupp.y/j N , then for every M 2 N we have ŒŒMy Œu,1/, x M C

N . u

The same assertion holds for functions x, y 2 .L1 C L1 /.0, 1/ (or x, y 2 L1.0, 1/). Proof. Define sequences ,  2 l1 by setting .n/ :D x..n C 1/u/ and .n/ :D y..n C 1/u/ for all n  0. We have .y Œu,1/ / .u / .y/  .x Œu,1// .u / .x/.

94

Chapter 3 Symmetric Operator Spaces

Hence, u  u  and, therefore,   . Clearly, jsupp./j .N  u/=u. By Lemma 3.4.8,

 N N u C1 M C . ŒŒM ,  M C u u On the other hand, it follows from Lemma 3.4.7 that ŒŒMy Œu,1/, x ŒŒMu , u  ŒŒM ,  M C

N . u

The next lemma, together with Lemma 3.4.9, provides a crucial technical estimate for Proposition 3.4.11. Lemma 3.4.10. Let x, y 2 l1 be positive sequences such that y C x. For  2 N as in Definition 3.4.1 and for every u  0, we have .y/ Œ2 u,1/ 

 .x/ Œu,1/ . 1

The same assertion holds for functions x, y 2 .L1 C L1 /.0, 1/ (or x, y 2 L1.0, 1/). Proof. For every t > 0, we have Z

uCt

Z .s, y/ds

u

Let

uCt

.s, x/ds

u

 Z AD t 0:

uCt

.s, y/ds 

u

t C .  1/u t

 1

Z

uCt

Z

uCt

.s, x/ds. u

 .s, x/ds .

u

For every t 2 A, we have t < .  1/2u. For every t  sup.A/, we have Z

uCt u

 .s, y/ds 1

Z

uCt

.s, x/ds.

(3.7)

u

Observe that A is a closed set and, therefore, sup.A/ 2 A. Thus, Z

uCsup.A/

.s, y/ds D

u

 1

Z

uCsup.A/

.s, x/ds.

(3.8)

u

Subtracting (3.8) from (3.7), we obtain Z

uCt

uCsup.A/

.s, y/ds

 1

Z

uCt

.s, x/ds. uCsup.A/

(3.9)

95

Section 3.4 Uniform Submajorization

It follows from (3.9) and the fact that sup.A/ < .  1/2 u that .y/ Œ2 u,1/ .y/ ŒuCsup.A/,1/ 

 .x/ ŒuCsup.A/,1/ 1

 .x/ Œu,1/ . 1

The following proposition connects uniform submajorization and the Minkowski functional for Q.x/, 0 x 2 l1 (whose value is provided in Lemma 3.4.5). Proposition 3.4.11. If x D .x/ 2 l1 and y D .y/ 2 l1 are such that y C x, then 1 lim ŒŒNy, x 1. N !1 N The same assertion holds for functions x, y 2 .L1 CL1 /.0, 1/ (or, x, y 2 L1.0, 1/). Proof. Let   3 be as in Definition 3.4.1. For 0 k < , set A0k D Œ0,  /, Ank D Œ 3k

B0k D Œ0, 3k /,

3kC3C3.n1/

,

3kC3n

Bnk D Œ3kC3.n1/ , 3kC3n/,

/,

n  1, n  1,

Ak D ZC D

1 [ nD0 1 [

Ank Bnk .

nD0

It follows from Lemma 3.4.10 (with u D 23kC3.n1/) that  x 3kC3.n1/ ,1/ .   1 Œ2

(3.10)

 x 3kC3.n1/ ,3kC3n / .   1 Œ2

(3.11)

y Œ23kC2C3.n1/,1/  Hence, y Œ23kC2C3.n1/,3kC3n / 

The left-hand side of (3.11) has a support of length at most 3kC3n. For every n  1, it follows from (3.11) and Lemma 3.4.9 (with u D 3kC3.n1/ ) that hh My Œ.22C1/3kC3.n1/ ,3kC3n / ,

ii  x Œ3kC3.n1/ ,3kC3n / M C 3. 1

Since   3 (in particular, 22 C 1 3), it follows that for n  1, we have hh My An,k ,

ii  x Bn,k M C 3. 1

The inequality (3.12) for n D 0 follows from Lemma 3.4.8.

(3.12)

96

Chapter 3 Symmetric Operator Spaces

It follows from (3.12), and Lemma 3.4.7, that hh My Ak ,

ii hh M ii M   x D x Bnk M C 3. (3.13) My Ank , 1 1 1

1

nD0

nD0

However, for every m > 0 there exists exactly one 0 k <  such that m … Ak . It follows that 1 X y Ak . y.  1/ C y Œ0,1/ D kD0

It follows from (3.13) that hh

ii 1 ii X hh   M.  1/y, My Ak , x x .M C 3 /. 1 1 kD0

Let M D 1 C ŒN =.  1/2 . It follows that ŒŒNy, x ŒŒM.  1/2 1y, x D .M C 23/ N.

hh

M.  1/y,

ii  x 1

 2 / C 23. 1

Letting N ! 1, we obtain lim sup N !1

  2 1 ŒŒNy, x . N 1

Since  can be chosen arbitrary large, the assertion follows. We are now ready to prove Theorem 3.4.2. Proof of Theorem 3.4.2. Let x, y 2 l1 be positive sequences. Without loss of generality, we assume that x D .x/ and y D .y/. (a) By assumption, there exist n 2 N, k  0 and zk , 1 k n, such that .zk / .x/ and such that n n X X k zk D y, k D 1. kD1

kD1

It follows from Lemma 3.4.4 that yC

n X kD1

Hence, y C x.

k .zk /

n X kD1

k .x/ D .x/.

Section 3.5 Symmetric Operator Spaces from Symmetric Function Spaces

97

(b) If y C x, then it follows from Proposition 3.4.11 that 1 ŒŒNy, x 1. N !1 N lim

By Lemma 3.4.5, the latter limit is the Minkowski functional for the convex hull of the set fz  0 : .z/ .x/g. The assertion now follows from the definition of the Minkowski functional. If x, y 2 .L1 C L1 /.0, 1/ or x, y 2 L1.0, 1/, then the argument remains the same.

3.5 Symmetric Operator Spaces from Symmetric Function Spaces The main aim of this section is to prove, using Theorem 3.4.2, that the operator space E.M, / :D fA 2 S.M, / : .A/ 2 Eg associated to a symmetric function space E is a symmetric operator space with norm kAkE :D k.A/kE ,

A 2 E.M, /.

First we show that E.M, / is normed. Corollary 3.5.1. Let M be a semifinite von Neumann algebra. Let E be a symmetric function space. It follows that the corresponding symmetric operator space E.M, / is a normed space. Proof. Let A, B 2 E.M, /. We show that the triangle inequality holds for k  kE . The other properties are evident. It follows from Lemma 2.3.15 that jA C Bj U  AU C V  BV for some partial isometries U , V 2 M. It follows from Lemma 3.4.4 that .A C B/ .U  AU C V  BV / C .A/ C .B/. By Corollary 3.4.3, kA C BkE k.A/kE C k.B/kE D kAkE C kBkE . We now show that E.M, / is complete. The proof is based on the fundamental estimate obtained in Lemma 3.5.4.

98

Chapter 3 Symmetric Operator Spaces

Lemma 3.5.2. Let M be a semifinite von Neumann algebra. Let Ak 2 .L1 C L P11/.M, /, k 2 N, be positive operators and let ˛k 2 RC , k 2 N, be such that kD1 ˛k 1. It follows that Z

b a

  X 1 1 Z X  s, Ak ds kD1

Here, the series

P1

kD1 Ak

kD1

b ˛k a

.s, Ak /ds,

0 < a < b.

is assumed to be convergent in .L1 C L1 /.M, /.

Proof. It follows from Lemma 3.3.5 that Z

a

  X 1 1 Z X  s, Ak ds 

0

kD1

kD1

˛k a

.s, Ak /ds.

(3.14)

0

On the other hand, we have Z

b

 X  1 1 Z X  s, Ak ds

0

kD1

kD1

b

.s, Ak /ds.

(3.15)

0

Subtracting (3.14) from (3.15), we obtain the assertion. Lemma 3.5.3. Let M be a semifinite von Neumann algebra. P Let Ak 2 .L1 C L1 /.M, /, k 2 N, and let ˛k 2 RC , k 2 N, be such that 1 kD1 ˛k 1. It follows that Z

b a

Here, the series

  X 1 1 Z X  s, Ak ds kD1

P1

kD1 Ak

kD1

b ˛k a

.s, Ak /ds,

0 < a < b.

is assumed to be convergent in .L1 C L1 /.M, /.

Proof. Fix n 2 N. By Lemma 2.3.15, there exist partial isometries Uk , 1 k n, such that n n ˇX ˇ X ˇ ˇ Ak ˇ Uk jAk jUk . ˇ kD1

kD1

It follows from Lemma 3.5.2 that Z

b a

  X n n Z X  s, Uk jAk jUk ds kD1

kD1

b ˛k a

.s, Uk jAk jUk /ds,

80 < a < b.

99

Section 3.5 Symmetric Operator Spaces from Symmetric Function Spaces

Therefore, Z

b a

  X n n Z X  s, Ak ds

Since the series Z

b

a

kD1

kD1

P1

kD1 Ak

˛k a

.s, Ak /ds,

80 < a < b.

is convergent in .L1 C L1 /.M, /, it follows that

 X  Z 1  s, Ak ds D lim

b

n!1 a

kD1

b

 X  n  s, Ak ds

n X

lim

n!1

Z

kD1

kD1 b

˛k a

.s, Ak /ds D

1 Z X kD1

b ˛k a

.s, Ak /ds

for every 0 < a < b. Lemma 3.5.4. Let M be a semifinite von Neumann algebra and let Ak 2 .L1 C L1 /.M, /. We have 1 1 X X Ak C 2 2k .Ak /. kD1

kD1

P Here, the series 1 kD1 Ak is assumed to be convergent in .L1 C L1 /.M, / and the P1 series kD1 2k .Ak / is assumed to be convergent in S (convergent in measure). Proof. Let 0 < 2a < b. It follows from Lemma 3.5.3 that  Z b  X 1 1 Z b X  s, Ak ds .s, Ak /ds. 2a

kD1

kD1

Set k D

2k b  2a

Z

21k a

2k b 21k a

8k 2 N.

.s, Ak /ds,

It is clear that .s, Ak / k for every s  2k b and .s, Ak /  k for every s 21k a. Therefore, Z

b 21k a

Z .s, Ak /ds

2k b 21k a

Z .s, Ak /ds C

b 2k b

k ds D .b  21k a/k

and Z

b a

.2k .Ak //.s/ds  2

k

Z

2k b

21k a

Z .s, Ak /ds C



21k a 2k a

k ds

 .b  a/k .

100

Chapter 3 Symmetric Operator Spaces

Therefore, Z

b 21k a

Z .s, Ak /ds .b  21k a/k 2.b  a/k 2

Hence,

Z

b 2a

  X Z 1  s, Ak ds 2 kD1

b a

X 1

b a

.2k .Ak //.s/ds.

 2k .Ak / .s/ds.

kD1

The next theorem is the main result of this chapter. It answers Question 2.5.4 in the affirmative for the atomless case, and it answers Question 2.5.5. Theorem 3.5.5. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . For every symmetric function space E on .0, 1/ if .1/ D 1 (or .0, 1/ if .1/ D 1), the set E.M, / :D fA 2 S.M, / : .A/ 2 Eg is a symmetric operator space when equipped with the norm kAkE :D k.A/kE , A 2 E.M, /. Proof. By Corollary 3.5.1, E.M, / is a normed space. We only have to prove completeness. For this purpose, fix a Cauchy sequence fAn gn2N in E.M, /. We will prove the existence of A 2 E.M, / such that An ! A in E.M, /. For every k > 0, there exists mk such that kAm  Amk kE 4k for m  mk . Set Bk D AmkC1  Amk . Clearly, kBk kE 4k for every k 2 N. In particP ular, kBk kL1CL1 4k and, therefore, the series 1 kD1 Bk converges in .L1 C L1 /.M, / by Example 2.6.7. It follows from Lemma 3.5.4 that 1 X

1 X

Bk C 2

kDn

2kC1n .Bk /,

n  3.

kDn

It now follows from Theorem II.4.5 of [139] that 1 X  .B / k 2kC1n kDn

E

1 X

2

2kC1n kBk kE 22n

kDn

Therefore, 1 X kDn

2kC1n .Bk / ! 0

1 X kDn

2k D 232n .

Section 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces

101

 P1 P1 in E. By Corollary 3.4.3,  ! 0 in E. By conk kDn B P1 2 E and . kDn Bk / P struction of E.M, / we have that kDn Bk 2 E.M, / and 1 kDn Bk ! 0 in E.M, /. The next section gives a similar theorem for the case of a symmetric sequence space.

3.6 Symmetric Function Spaces from Symmetric Sequence Spaces Theorem 3.5.5 shows that a symmetric function space on the semi-axis defines a symmetric operator space on a semifinite von Neumann algebra M. In this section we provide the counterpart for a symmetric sequence space. We do so by associating a symmetric function space on .0, 1/ to a symmetric sequence space E. Let A D fAk g be a (finite or infinite) sequence of disjoint sets of finite measure and denote by A the collection of all such sequences. We need the notion of an expectation operator (see [20]). Definition 3.6.1. The expectation operator E.jA/ : .L1 C L1 /.0, 1/ ! .L1 C L1 /.0, 1/ (or E.jA/ : L1.0, 1/ ! L1 .0, 1/) is defined by setting  X 1 Z x.s/ds Ak . E.xjA/ :D m.Ak / Ak k

Lemma 3.6.2. For every x 2 .L1 C L1 /.0, 1/ and for every A 2 A, we have E.xjA/  x. Proof. Every expectation operator is a contraction in both L1 and L1 . The assertion now follows from Theorem II.3.4 of [139]. In this section, A D f.n  1, ngn2N is a partition of the semi-axis. Clearly, E.jA/ maps .L1 C L1 /.0, 1/ into the set of step functions which can be identified with sequences. Construction 3.6.3. Let E be a symmetric sequence space. Let F be the linear space of all x 2 .L1 C L1 /.0, 1/ for which E..x/jA/ 2 E. The main aim of this section is to introduce a Banach norm on the function space F so that F becomes a symmetric function space on the semi-axis. Lemma 3.6.4. Let M be an atomless semifinite von Neumann algebra. If A, B 2 .L1 C L1 /.M, / are positive operators, then E..A C B/jA/ C E..A/ C .B/jA/ C 21=2 E..A C B/jA/.

102

Chapter 3 Symmetric Operator Spaces

Proof. We only prove the left-hand side inequality. Proof of the right-hand side inequality is identical. If b is integer, then it follows from Lemma 3.3.3 that Z b Z b E..A C B/jA/.s/ds D .s, A C B/ds 0

0

Z

0 b

Z D

b

..s, A/ C .s, B//ds

E..A/ C .B/jA/.s/ds.

0

However, the inequality Z Z b E..A C B/jA/.s/ds 0

b

E..A/ C .B/jA/.s/ds

(3.16)

0

is piecewise linear and, therefore, holds for every b > 0. If 2a is integer, then it follows from Lemma 3.3.5 that Z 2a Z 2a E..A C B/jA/.s/ds D .s, A C B/ds 0 0 Z a ..s, A/ C .s, B//ds  0 Z a  E..A/ C .B/jA/.s/ds. 0

However, the inequality Z Z 2a E..A C B/jA/.s/ds  0

a

E..A/ C .B/jA/.s/ds

(3.17)

0

is piecewise linear and, therefore, holds for every a > 0. Subtracting this inequality, we obtain Z b Z b E..A C B/jA/.s/ds E..A/ C .B/jA/.s/ds (3.18) a

2a

Corollary 3.6.5. The function space F given in Construction 3.6.3 admits a norm defined by setting kxkF :D kE..x/jA/kE , x 2 F . Proof. It follows from Lemma 3.6.4 that E..x C y/jA/ C E..x/ C .y/jA/

Section 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces

103

provided that x, y 2 .L1 C L1 /.0, 1/ are positive functions. By Theorem 3.4.2 and the triangle inequality in E, we have kE..x C y/jA/kE kE..x/jA/kE C kE..y/jA/kE . Theorem 3.6.6. The function space F equipped with the norm in Corollary 3.6.5 is a symmetric function space. Proof. The norm k kF is already symmetric. We only have to prove the completeness of .F , k  kF /. Fix a Cauchy sequence xm 2 F , m  0. By Example 2.6.7, xm , m  0, is a Cauchy sequence in .L1 C L1 /.0, 1/ and therefore xm ! x in .L1 C L1 /.0, 1/ for some x 2 .L1 C L1 /.0, 1/. For every k  0, there exists mk such that kxm  xmk kF 4k for m  mk . Set k y Pk 1D xmkC1  xmk . Clearly, kyk kF 4 for every k  0. In particular, the series kDn yk converges to x  xmn in .L1 C L1 /.0, 1/ for every n  1. We have kE..yk /jA/kE 4k . By Theorem II.4.5 of [139], we have k2k E..yk /jA/kE 2k . Therefore, k

1 X

2k E..yk /jA/kE

kDn

1 X

2k D 21n .

(3.19)

kDn

On the other hand, we have  X X 1 1 2k .yk /jA D 2k E..yk /jA/. E kDn

kDn

It follows from the preceding formula and the definition of norm in F that 1 X 2k .yk /

F

kDn

1 X D 2k E..yk /jA/ kDn

E

21n .

It follows from Lemma 3.5.4 and Corollary 3.4.3 that 1 X yk kDn

F

1 X 2 2k .yk / kDn

F

22n .

(3.20)

Thus, the subsequence xmn converges to x in F . Hence, the sequence xn converges to x in F . We can associate to the symmetric sequence space E (through the symmetric function space F ) a symmetric operator space.

104

Chapter 3 Symmetric Operator Spaces

Corollary 3.6.7. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . For every symmetric sequence space E, the set E.M, / :D fA 2 S.M, / : E..A/jA/ 2 Eg is a symmetric operator space when equipped with the norm kAkE :D kE..A/jA/kE , A 2 E.M, /. Proof. We apply Theorem 3.5.5 to the symmetric function space F in Theorem 3.6.6. The assertion follows. If M is atomic (recall from Section 2.3 that the standing assumption on an atomic semifinite von Neumann algebra is that .p/ D 1 for any non-zero minimal projection p 2 Proj.M/), then E..A/jA/ D .A/ for any A 2 S.M, /. Corollary 3.6.7 answers Question 2.5.4 for the atomic case.

3.7 Notes Hardy–Littlewood Submajorization The text already mentioned the foundational book of Hardy, Littlewood and Polya [109] and the more modern book by A. W. Marshall and I. Olkin [161]. For applications of submajorization theory to the interpolation theory of linear operators, see [10, 139]. Lemma 3.3.7 is a slight generalization of Lemma 2.3 from [40]. The combined result given in Theorems 3.3.3 and 3.3.4 can be found in [103, Proposition 1.10 (i)]. Completeness Solving Question 2.5.4 had been a long standing goal of the second named author. In fact, Theorem 3.1.1 appeared in the paper [129] as a combination of [129, Theorem 8.7] and [129, Theorem 8.11]. These results generalize a number of earlier results in the literature, in particular, where the completeness of symmetric operator spaces was established under certain additional conditions on the (quasi-)norm k  kE (see e.g. [262], [264, Proposition 2.8 (ii)], [232, Corollary 1], [233, Theorem 1.2.4], [260, Lemma 4.1], [69, Theorem 4.5], [71, Corollary 2.4]). We refer the reader to the forthcoming paper [229] for the affirmative solution for quasi-normed symmetric operator spaces. Uniform Submajorization Uniform submajorization, as a means to the solution of Question 2.5.4, was introduced in [129]. Theorem 3.4.2 is a special case of a stronger result obtained in [129]. See also the comments in [183]. For the historical background to the paper [129] we quote from a memorial written by the second named author; reprinted with permission from the Nigel Kalton Memorial Website1. In 1998, together with B. de Pagter, P. Dodds and E. Semenov, we approached the notion of a Dixmier trace, which we called, at that time, a symmetric functional, as a part of a general theory of singular traces on symmetrically normed 1

http://kaltonmemorial.missouri.edu/

Section 3.7 Notes operator ideals of L.H / (and, more generally, on symmetrically normed operator bimodules on semifinite von Neumann algebras). At that time, I was vaguely aware of earlier work of Nigel with Figiel and with Dykema. However, their framework was subtly different and I could not immediately see its implications for our own work. [A] meeting in Oberwolfach [in 2004] was a perfect chance to tell to Nigel about our results and also state a number of problems some of which we tried ourselves and some of which were fairly new and which I thought of as important from the viewpoint of noncommutative geometry. Somewhat incredibly, Nigel was interested and the very next day he approached me with a tentative solution to the problem which we thought was very hard. Nigel suggested a way how one can attempt to construct symmetric functionals (and unitarily invariant linear functionals on a special Lorentz ideal of importance in noncommutative geometry) which were not monotone with respect to the Hardy–Littlewood [sub]majorization. The idea was so nice that I liked it straight away. The problem was with its technical implementation, or rather with my understanding of the latter. It was some time before I could present a strict and complete technical record of Nigel’s idea, which was initially stated in two or three lines on a table napkin. Indeed, many times during my subsequent collaboration with Nigel, I noted that he didn’t seem to experience the technical difficulties which would set back anybody else. Back in 2004, we parted with the agreement that I would try to continue working with the argument and see for what class of Lorentz spaces it is applicable. That took a few years (the paper was only published in 2008) and a few meetings in Adelaide and Missouri which still are (and forever will be) a treasured part of my memory. Our further collaboration was firmly centered on the theory of singular traces. I was able to contribute to our work my knowledge of (sometimes obscure) works on the theory of symmetric spaces from the former Soviet Union (Braverman, Mekler, Russu, Sedaev); however, the earlier ideas and techniques of Nigel (from his papers with Figiel and Dykema, which I mentioned earlier) have come to play an essential role in our approach. It was an absolute pleasure for me to see how ideas and techniques born from completely different perspectives (and motivations) became central to the study of singular traces and their applications in noncommutative geometry. Eventually, it had become clear to us that there exists a single thread which permeates works on commutators, unitary orbits, various geometrical questions in the theory of symmetric operator spaces and noncommutative geometry which was both fascinating and fruitful. Of course, this realization would never have happened if Nigel was not the mathematical giant that he was. However, our main achievement with Nigel belongs to a different area (even though its applications, by now, have proved paramount for singular trace theory as well). That main achievement is an infinite-dimensional analogue of a finite-dimensional result of John von Neumann from 1937 and the new notion which was invented to obtain this analogue. In that remarkable paper, von Neumann laid the foundation of what was later to become the theory of symmetrically normed (or unitarily-invariant) operator ideals. It is also the first paper where the so-called “noncommutative Lp -spaces” made their appearance. I read this pa-

105

106

Chapter 3 Symmetric Operator Spaces

per of von Neumann as a very young man and realized the beauty of its ideas and noted that it suggested immediately a number of infinite-dimensional questions which were to occupy me for the next 25 years. One of these problems which I tried to resolve in my PhD thesis (1988) was whether a positive unitarilyinvariant functional on a given unitarily-invariant ideal E in the algebra L.H / of all bounded linear operators on a Hilbert space H is a (Banach) norm provided this is the case for its restriction to the diagonal subspace of E (that is, on the set of all operators from E which are diagonal with respect to a given orthonormal basis in H ). In my PhD thesis, I answered this question under the additional assumption that the norm is monotone with respect to Hardy-Littlewood [sub]majorization. That was a frustrating restriction and, for many years, I returned again and again to that problem. It was only natural that at some stage in 2006 while we were finalizing our work (begun earlier in Oberwolfach) on singular traces which are not monotone with respect to the Hardy–Littlewood [sub]majorization, I again looked at this infinite-dimensional analogue of von Neumann’s result and explained it to Nigel together with a rather long account of various approaches I had tried in the past. Basically, it took Nigel a couple of weeks and a long flight from Adelaide to the US to come up with an outstanding new idea which we later termed “uniform Hardy–Littlewood [sub]majorization”. This was the key to the solution of that problem. The paper entitled “Symmetric norms and spaces of operators” was published in 2008 by Crelle’s journal and we both believed that this paper will prove itself useful for various questions in analysis of symmetric spaces. One must also bear in mind the extraordinary reticence and modesty (typical for Nigel) and his extreme aversion to boasting. His modesty was as great as his genius. It took some effort from my part to submit that article to a “big journal” and he acquiesced only because we both thought of it as a major contribution to the area.

Chapter 4

Symmetric Functionals

4.1 Introduction In Section 2.7 we introduced the notion of a symmetric functional on a symmetric operator space. We repeat it here for convenience. Definition 4.1.1. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . Let E.M, / be a symmetric operator space. The functional ' 2 E.M, / is called (a) symmetric if '.A/ D '.B/ whenever .A/ D .B/ for 0 A, B 2 E.M, /. (b) fully symmetric if '.B/ '.A/ whenever .A/  .B/ for 0 A, B 2 E.M, /. The adjective "symmetric" applied to a functional on E.M, / implies automatically that the functional is a continuous linear functional on E.M, /. Another aspect to note from the definition is that a symmetric functional is not necessarily a positive linear functional whereas a fully symmetric functional is automatically a positive linear functional. Lemma 2.7.4 of Chapter 2 noted that the symmetric functionals describe the continuous traces on a symmetric operator space that can be constructed using a formula on singular values. When M is an atomless (or atomic) factor all continuous traces on E.M, / are symmetric functionals. In particular, when M D L.H / for a separable Hilbert space H , the theory that we present in this chapter accounts for all continuous traces on symmetrically normed ideals of compact operators. Chapter 3 states that the Calkin correspondence is a functor from symmetric function spaces to symmetric operator spaces, which is bijective in the atomless (or atomic) case. It is then natural to ask whether the Calkin correspondence extends to a functor from symmetric functionals on symmetric function spaces to symmetric functionals on symmetric operators spaces (in different terminology, do the symmetric functionals on a function space lift to the corresponding operator space). In particular, if ' 2 E  is a symmetric functional on the symmetric function space E, and E.M, / is the symmetric operator space corresponding to E in Theorem 3.5.5, does the assignment L.'/.A/ :D '..A//,

0 A 2 E.M, /

define a symmetric functional on E.M, /, and does the assignment provide a bijective correspondence in the atomless (or atomic) case? The affirmative answer is provided

108

Chapter 4 Symmetric Functionals

in Theorem 4.4.1 in Section 4.4. A similar functor from symmetric functionals on symmetric sequence spaces to symmetric functionals on symmetric operator spaces is shown using the construction of Section 3.6. The importance of the functors is that we can answer the following natural existence questions about symmetric functionals on operator spaces by answering the same questions about symmetric functionals on symmetric function (or sequence) spaces. Question 4.1.2. (a) Which symmetric operator spaces admit a nontrivial symmetric functional? (b) Which fully symmetric operator spaces admit a nontrivial fully symmetric functional? (c) Which fully symmetric operator spaces admit a nontrivial symmetric functional that is not a fully symmetric functional? The crucial component in answering these questions is the Figiel–Kalton theorem concerning the notion of the center of a symmetric function space (and the tool of uniform submajorization introduced in the last chapter). In answering Question 4.1.2 we investigate and describe for the atomless (or atomic) case the class of symmetric operator spaces admitting a symmetric functional. We consider how many symmetrical functionals may be admitted. We show that a symmetric functional decomposes into two Hermitian (or four positive) symmetric functionals and that the set of Hermitian symmetric functionals form a Banach lattice which is either trivial, 1-dimensional or infinite-dimensional. Another result shown in this chapter is that symmetric functionals decompose into normal and singular parts. It is shown that if E.M, / 6 L1.M, / (we are still considering here that M is atomless or atomic), then every symmetric functional ' on E.M, / is singular. To be mundane ' D 0 C 's where 's is a singular symmetric functional and 0 is the trivial normal functional. If E.M, /  L1 .M, / then ' D ˛ C 's for some constant ˛ and a singular symmetric functional 's . So the theory of singular symmetric functionals in the atomless (or atomic) case is the entire theory of symmetric functionals beyond the trivial or canonical normal trace. The following theorem is the main result of this chapter. It yields a complete answer to Question 4.1.2. For brevity, we state the result for the case M D L.H / and in terms of continuous traces (knowing, from Lemma 2.7.4, that all continuous traces

Section 4.2 Jordan Decomposition of Symmetric Functionals

109

are symmetric functionals for the factor L.H /). The statement (and the proof) for semifinite atomless or atomic von Neumann algebras can be found in Section 4.9. Theorem 4.1.3. Let E be a symmetric ideal of L.H /. Consider the following conditions. (a) There exist nontrivial continuous singular traces on E. (b) There exist nontrivial continuous singular traces on E, which are fully symmetric. (c) There exist nontrivial continuous singular traces on E, which are not fully symmetric. (d) E ¤ L1 and there exists an operator A 2 E such that lim

m!1

(i)

1 k A ˚    ˚ A kE > 0. m „ ƒ‚ … m times

(4.1)

The conditions (a) and (d) are equivalent for every symmetric ideal.

(ii) The conditions (a), (b) and (d) are equivalent for every symmetric ideal that is fully symmetric. (iii) The conditions (a)–(d) are equivalent for every symmetric ideal that is fully symmetric and equipped with a Fatou norm. Observe that if E D L1 , then the condition (4.1) trivially holds for every 0 ¤ A 2 L1 . The ideal of trace class operators L1 admits the continuous trace Tr, so we have the result that a symmetric ideal of compact operators admits a continuous trace if and only if (4.1) holds (see Corollary 4.9.3).

4.2 Jordan Decomposition of Symmetric Functionals Let M be a von Neumann algebra equipped with a fixed faithful normal semifinite trace  and let E.M, / be a symmetric operator space. A functional ' 2 E.M, / is called Hermitian if '.A/ D '.A / for every A 2 E.M, /. Definition 4.2.1. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. The mapping 'C : E.M, /C ! R, defined by setting 'C .A/ :D supf'.B/ : 0 B Ag, is called the positive part of the functional '.

0 A 2 E.M, /,

110

Chapter 4 Symmetric Functionals

We show below (in Proposition 4.2.8) that the linear extension of 'C is a positive symmetric functional on E.M, /. The main result of this section is the following analog of the Jordan decomposition of a normal form on a von Neumann algebra (see e.g. Theorem 5.17 in [227]). The reader should bear in mind that symmetric functionals are generally singular and, therefore, the classical techniques used on normal functionals are not applicable here. Theorem 4.2.2. Let M be an atomless or atomic semifinite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a symmetric functional. There exist positive symmetric functionals 'k 2 E.M, / , 1 k 4, such that ' D '1  '2 C i'3  i'4 . The functionals '1  '2 and '3  '4 are Hermitian with positive parts '1 and '3 , respectively. Proof. Let ' be a symmetric functional on E.M, /. The mapping 'N : A ! '.A / is also a symmetric functional. The equality 'D

1 1 .' C '/ N C i .'  '/ N 2 2i

is the decomposition of a symmetric functional ' into Hermitian symmetric functionals. We can therefore assume, without loss of generality, that ' is Hermitian. Given that the linear extension of 'C is a positive symmetric functional on E.M, / (see Proposition 4.2.8 below), we define a functional ' 2 E.M, / by setting ' :D 'C  '. Since 'C .A/  '.A/ for every positive operator A 2 E.M, /, it follows that ' is also a positive symmetric functional. The rest of this section proves Proposition 4.2.8. Lemma 4.2.3. Let M be a semifinite von Neumann algebra with .1/ D 1, and let E.M, / be a symmetric operator space such that M  E.M, /. Every continuous symmetric functional ' on E.M, / vanishes on M. Proof. Let q 2 M be a projection such that .q/ D .1  q/ D 1. We have .q/ D .1  q/ D 1 and, therefore, '.q/ D '.1  q/ D '.1/. It follows that '.1/ D 0. Let p 2 M be a projection. If .p/ D 1, then .p/ D 1 and, therefore, '.p/ D '.1/ D 0. If .p/ < 1, then .1  p/ D 1 and, therefore, '.1  p/ D '.1/ and '.p/ D 0. Hence, '.p/ D 0 in either case. Recall that every element in M can be uniformly approximated by linear combinations of projections. If M  E.M, /, then it follows from the Closed Graph Theorem

Section 4.2 Jordan Decomposition of Symmetric Functionals

111

that k  kE k  k1 . Thus, every element in M can be approximated by linear combinations of projections in the norm topology of E.M, /. It now follows from the previous paragraph that the bounded symmetric functional ' vanishes on M. In the next lemma s , s > 0, is the dilation operator on .L1 C L1 /.0, 1/. Lemma 4.2.4. Let M be an atomless or atomic semifinite von Neumann algebra and let E.M, / be a symmetric operator space. Let ' be a symmetric functional on E.M, / and let A1 , A2 2 E.M, / be positive operators. (a) If .A2 / D 1=n .A1 /, n 2 N, then '.A2 / D 1=n'.A1 /. (b) If .1/ D 1 and .A2 / D n .A1 /, n 2 N, then '.A2 / D n'.A1 /. is well Proof. Suppose first that .A2 / D 1=n .A1 /, n 2 N. The operator A˚n 2 defined and .A1 / D .A˚n /. On the other hand, 2 A˚n D A2 ˚ 0    ˚ 0 C 0 ˚ A2 ˚ 0    ˚ 0 C    0 ˚ 0 ˚    A2 . 2 Since ' is a symmetric functional, it follows that '.A˚n 2 / D n'.A2 /. Again using the fact that ' is symmetric, we obtain that '.A1 / D n'.A2 /. This proves the first assertion. The second assertion now follows from the fact that 1=n n is the identity operator on l1 (or L1 .0, 1/). The following lemma shows that one can replace the standard order in Definition 4.2.1 with uniform submajorization. This lemma demonstrates the key role played by uniform submajorization for symmetric functionals. Lemma 4.2.5. Let M be an atomless or atomic semifinite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. We have 'C .A/ D supf'.B/ : 0 B C Ag,

0 A 2 E.M, /.

Proof. Let 0 B be such that B C A. It follows from Corollary 3.4.3 that B 2 E.M, /. Set D D .B  .1, B//C . Since 0 D B and 0 B C A, it follows that 0 D C A and .1, D/ D 0. We claim that '.B/ D '.D/. If the algebra M is finite or if M 6 E.M, /, then .1, B/ D 0. In this case, B D D and the claim follows immediately. Now let M be an infinite von Neumann algebra and let E.M, /  M. By Lemma 4.2.3, ' vanishes on M. Since B  D 2 M, the claim follows.

112

Chapter 4 Symmetric Functionals

Fix  > 0. It follows from Theorem 3.4.2 that there exist positive functions zk , 1 k n, and positive constants k , 1 k n, such that .zk / .A/ and .1  /.D/ D

n X kD1

k zk ,

n X

k D 1.

(4.2)

kD1

Let i be the mapping constructed in Theorem 2.3.11. Set Dk D i.zk / for 1 k n. Since .zk / .A/ and .Dk / D .zk /, 1 k n, it follows that 0 Dk and .Dk / .A/ P for 1 k n. Applying i to both sides of (4.2), we obtain that .1  /D D nkD1 k Dk . Therefore, .1  /'.B/ D .1  /'.D/ D

n X

k '.Dk / supf'.B/ : 0 B Ag.

kD1

Since  > 0 is arbitrarily small, the assertion follows. The following lemma is the first step in showing that the linear extension of 'C is a positive linear functional on E.M, /. Lemma 4.2.6. Let M be an atomless or atomic semifinite von Neumann algebra and let A1 , A2 2 E.M, / be positive operators. If .A2 / D 21=2.A1 /, then 'C .A1 / D 'C .A2 /. Proof. First, we prove that 'C .A1 / 'C .A2 /. By the assumption of the lemma, A1 C A2 . It follows from Lemma 4.2.5 that 'C .A1 / D supf'.B/ : 0 B C A1 g supf'.B/ : 0 B C A2 g D 'C .A2 /. In order to prove the converse inequality, select 0 B A2 . Select a positive operator 0 C A1 such that .B/ D 21=2 .C /. It follows from Lemma 4.2.4 that '.B/ D '.C / 'C .A1 /. Taking the supremum over B, we obtain 'C .A2 / 'C .A1 /. The next lemma is the second step. Lemma 4.2.7. Let M be an atomless or atomic semifinite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. For all positive operators A1 , A2 2 E.M, /, we have 'C .A1 ˚ A2 / D 'C .A1 / C 'C .A2 / provided that A1 ˚ A2 2 E.M, /.

Section 4.2 Jordan Decomposition of Symmetric Functionals

113

Proof. If 0 B1 A1 and 0 B2 A2 , then 0 B1 ˚ B2 A1 ˚ A2 . Hence, supf'.B/ : 0 B A1 ˚ A2 g  supf'.B1 ˚ B2/ : 0 B1 A1 , 0 B2 A2 g and, therefore, 'C .A1 ˚ A2 /  'C .A1 / C 'C .A2 /.

(4.3)

In order to prove the converse inequality, let 0 B A1 ˚A2 . Let p be the support projection of A1 ˚ 0 and let U D p C i.1  p/. Since U is unitary, it follows that '.C / D '.U 1 C U / for every operator C 2 E.M, /. Note that Bp  U 1 .Bp/U D Bp  U 1 Bp D .1 C i/.1  p/Bp. Therefore, '..1p/Bp/ D 0 and, since ' is Hermitian, it follows that '.pB.1p// D 0. Hence, '.B/ D '.pBp/ C '..1  p/B.1  p//. On the other hand, we have pBp p.A1 ˚A2 /p D A1 ˚0,

.1p/B.1p/ .1p/.A1 ˚A2 /.1p/ D 0˚A2 .

Setting B1 :D pBp and B2 :D .1  p/B.1  p/, we have '.B/ D '.B1 C B2 / and 0 B1 A1 ˚ 0, 0 B2 0 ˚ A2 . Therefore, 'C .A/ D supf'.B/ : 0 B A1 ˚ A2 g supf'.B1/ C '.B2 / : 0 B1 A1 ˚ 0, 0 B2 0 ˚ A2 g. Thus, 'C .A1 ˚ A2 /  'C .A1 / C 'C .A2 /.

(4.4)

The assertion follows from (4.3) and (4.4). We are now in a position to prove that 'C is additive on the positive part of E.M, / and extends to a symmetric functional. Proposition 4.2.8. Let M be an atomless or atomic semifinite von Neumann algebra. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. For positive operators A1 , A2 2 E.M, /, we have 'C .A1 C A2 / D 'C .A1 / C 'C .A2 / and 'C .A1 / D 'C .A2 / if

.A1 / D .A2 /.

In particular, 'C extends to a positive symmetric functional on E.M, /.

114

Chapter 4 Symmetric Functionals

Proof. If 0 A1 , A2 2 E.M, /, then also A1 ˚ A2 2 E.M, /. Repeating the argument in Lemma 3.4.4, we obtain A1 ˚ A2 C A1 C A2 C 21=2 .A1 ˚ A2 /.

(4.5)

It follows from Lemma 4.2.5 and (4.5) that 'C .A1 ˚ A2 / D supf'.B/ : 0 B C A1 ˚ A2 g supf'.B/ : 0 B C A1 C A2 g D 'C .A1 C A2 /. It follows from Lemma 4.2.5 and (4.5) that 'C .A1 C A2 / D supf'.B/ : 0 B C A1 C A2 g supf'.B/ : 0 B C 21=2 .A1 ˚ A2 /g. Lemma 4.2.6 now implies that 'C .A1 C A2 / supf'.B/ : 0 B C A1 ˚ A2 g D 'C .A1 ˚ A2 /. We have shown that 'C .A1 C A2 / D 'C .A1 ˚ A2 / and now, applying Lemma 4.2.7, we obtain 'C .A1 C A2 / D 'C .A1 ˚ A2 / D 'C .A1 / C 'C .A2 /. Finally, if .A1 / D .A2 /, then the sets f0 B C A1 g and f0 B C A2 g are identical. Therefore, from Lemma 4.2.5 we obtain the equality 'C .A1 / D 'C .A2 /.

4.3 Lattice Structure on the Set of Symmetric Functionals Knowing that a Hermitian symmetric functional decomposes into positive symmetric functionals allows us to establish an interesting result (actually, from our viewpoint we view it as a remarkable result) that the set of all Hermitian symmetric functionals on a symmetric operator space E.M, / is a Banach lattice with respect to the operations _, ^ defined below. Later, in Theorem 4.10.1, we show that this Banach lattice is either trivial, 1-dimensional or infinite-dimensional. Definition 4.3.1. Let E.M, / be a symmetric operator space and let '1 , '2 2 E.M, / be Hermitian symmetric functionals. The mappings '1 _ '2 : E.M, /C ! R and '1 ^ '2 : E.M, /C ! R are defined by setting .'1 _ '2 /.A/ :D supf'1 .B/ C '2 .C / : 0 B, C , A D B C C g, 0 A 2 E.M, /. .'1 ^ '2 /.A/ :D inff'1 .B/ C '2 .C / : 0 B, C , A D B C C g, 0 A 2 E.M, /.

Section 4.3 Lattice Structure on the Set of Symmetric Functionals

115

Observe that ' _ 0 is the functional 'C from Definition 4.2.1. The following lemma extends Proposition 4.2.8. Lemma 4.3.2. Let E.M, / be a symmetric operator space and let '1 , '2 2 E.M, / be Hermitian symmetric functionals on E.M, /. The mappings '1 _ '2 and '1 ^ '2 defined in Definition 4.3.1 extend to symmetric functionals on E.M, /. Proof. For every positive A, we have .'1 _ '2 /.A/ D supf'1 .B/ C '2 .C / : 0 B, C , A D B C C g D supf'1 .A/ C .'2  '1 /.C / : 0 B, C , A D B C C g D '1 .A/ C supf.'2  '1 /.C / : 0 C Ag D '1 .A/ C .'2  '1 /C .A/. It follows from Proposition 4.2.8 that '1 _ '2 extends to a symmetric functional on E.M, /C . By Definition 4.3.1 we have '1 ^ '2 D ..'1 / _ .'2 //. Therefore, '1 ^ '2 is a symmetric functional. Corollary 4.3.3. Let E.M, / be a symmetric operator space. For every Hermitian symmetric functional ' 2 E.M, / , the mapping j'j :D ' _ .'/ is a positive symmetric functional on E.M, /. Proof. By Lemma 4.3.2, j'j is a symmetric functional on E.M, /. For every A  0, let B D A and C D 0 if '.A/  0 and let B D 0 and C D A otherwise. We have j'j.A/  '.B/  '.C / D j'.A/j  0. Hence, j'j is a positive functional. We require the norm   E  on E.M, / defined by setting 'E  :D

sup

ADA ,kAkE 1

j'.A/j,

' 2 E.M, / .

We have 'E  k'kE  2  'E  ,

' 2 E.M, / ,

and, therefore, the norms   E  and k  kE  are equivalent.

(4.6)

116

Chapter 4 Symmetric Functionals

Lemma 4.3.4. Let E.M, / be a symmetric operator space and let ' 2 E.M, / be a Hermitian symmetric functional. If   E  is the norm defined in (4.6), then 'E  D j'j E  . Proof. Since ' is Hermitian, it follows that 'E  D

sup

ADA ,kAkE 1

j'.A/j D

Therefore,

sup

ADA,kAkE 1

 '

E

D

sup

D0,kDkE 1

sup

ADA,jAjDD

'.A/.

 '.A/ .

On the other hand, we have sup

ADA ,jAjDD

'.A/ D supf'.AC  A / : AC , A  0, AC C A D Dg D j'j.D/.

It follows that 'E  D

sup

D0,kDkE 1

j'j.D/ D j'j E  .

Theorem 4.3.5. Let E.M, / be a symmetric operator space. The set of all Hermitian symmetric functionals in E.M, / is a Banach lattice with respect to the operations _ and ^ and the norm   E  . Proof. Clearly, the set of all Hermitian symmetric functionals on E.M, / is a closed subspace of E.M, / and is, therefore, a Banach space. First, we show that the operations _ and ^ are compatible with the natural order on the Hermitian part of E.M, / . It follows immediately from the definition of '1 _'2 that '1 _'2  '1 and '1 _'2  '2 . Conversely, let '  '1 and '  '2 . For every positive A 2 E.M, / and every decomposition A D B C C with positive B, C , we have '.A/ D '.B/ C '.C /  '1 .B/ C '2 .C /. Taking the supremum over all such decompositions, we obtain '  '1 _ '2 . Recall that j'j D ' _ .'/. If '1 , '2 2 E.M, / are Hermitian symmetric functionals such that j'2 j j'1 j, then it follows from Lemma 4.3.4 that '2 E  D j'2 jE  j'1 jE  D '1 E  . Note that the Hermitian part of the Banach dual E.M, / is not a lattice in general. Example 4.3.6. There exists a linear Hermitian functional ' 2 M2 .C/ such that 'C is nonlinear.

Section 4.4 Lifting of Symmetric Functionals

Proof. Let

117

      1 0 1 1 1 2 AD ,B D ,C D . 0 0 1 1 2 4

The matrices A and B=2 are projections. Hence, 0 A1 A implies that A1 D A,  2 Œ0, 1. Therefore, 'C .A/ D maxf'.A/, 0g. Similarly, 'C .B/ D maxf'.B/, 0g. Clearly, 10.A C B/  C and, therefore, 'C .A C B/  1=10'.C /. Setting   a11 a12 ! a22  a11 , ': a21 a22 we obtain that 'C .A/ D 'C .B/ D 0 and 'C .A C B/  3=10.

4.4 Lifting of Symmetric Functionals In this section, we explain how a symmetric functional on a symmetric function (or sequence) space lifts to a symmetric functional on the corresponding symmetric operator space. In the atomless (or atomic) case we show that the lift is a bijection. The main result is Theorem 4.4.1 below. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . If E is a symmetric function space, the corresponding symmetric operator space E.M, / given in Theorem 3.1.2 is E.M, / D fA 2 S.M, / : .A/ 2 Eg , kAkE D k.A/kE . If E is a symmetric sequence space, the corresponding symmetric operator space E.M, / given in Corollary 3.6.7 is E.M, / D fA 2 S.M, / : E..A/jA/ 2 Eg , kAkE D kE..A/jA/kE . Here E is the expectation operator defined in Section 3.6 and A :D f.n  1, ngn2N. We reiterate our convention that an atomic semifinite von Neumann algebra satisfies the condition that .p/ D 1 for any non-zero minimal projection p 2 Proj.M/, and that, in this case, E..A/jA/ D .A/. Theorem 4.4.1. Let E be a symmetric function (respectively, sequence) space and let E.M, / be the corresponding symmetric operator space as above. (a) If ' is a symmetric functional on E, then there exists a symmetric functional L.'/ on E.M, / such that L.'/.A/ D '..A//

118

Chapter 4 Symmetric Functionals

(respectively, L.'/.A/ D '.E..A/jA/ / for all positive A 2 E.M, /. The functional L.'/ is positive if the functional ' is positive. (b) Let M be an atomless (or atomic) semifinite von Neumann algebra. If ' is a symmetric functional on E.M, /, then there exists a symmetric functional L1 .'/ on E such that '.A/ D L1 .'/.x/ for all positive x 2 E and A 2 E.M, / such that .A/ D .x/. The functional L.'/ is positive if and only if the functional ' is positive. The theorem is proved below. Observe that the next corollary follows immediately from Theorem 4.4.1. Corollary 4.4.2. Let E and E.M, / be as in Theorem 4.4.1. (a) The functional L.'/ is a fully symmetric functional on E.M, / if ' is a fully symmetric functional on E. (b) If M is atomless (or atomic), then L.'/ is a fully symmetric functional on E.M, / if and only if ' is a fully symmetric functional on E. We now prove Theorem 4.4.1. We start with an intermediate result from Section 4.2. Lemma 4.4.3. Let E be a symmetric function (or sequence) space and let ' be a positive symmetric functional on E. (a) If x, y 2 EC are such that y C x, then '.y/ '.x/. (b) If x 2 E then '..x// D '.21=2 .x//. Proof. The first part is immediate from Lemma 4.2.5 since '.y/ 'C .x/ D '.x/. The second part is Lemma 4.2.4. We can lift symmetric functionals between a symmetric sequence space E and the symmetric function space F :D fx 2 .L1 C L1 /.0, 1/ : E..x/jA/ 2 Eg , kxkF :D kE..x/jA/kE constructed in Section 3.6. We identify the sequence space E with the subspace in F spanned by the characteristic functions of the partition A D f.n  1, ngn2N .

119

Section 4.4 Lifting of Symmetric Functionals

Theorem 4.4.4. Let E be a symmetric sequence space and let F be the symmetric function space as above. (a) Every symmetric functional on E extends to a symmetric functional on F . (b) The restriction of every symmetric functional on F to E is a symmetric functional. Proof. Let ' be a positive symmetric functional on E. Set '.x/ D '.E..x/jA//,

0 x 2 F.

If 0 x, y 2 F , then it follows from Lemma 3.6.4 that E..x C y/jA/ C E..x/ C .y/jA/ C 21=2E..x C y/jA/. It follows from Lemma 4.4.3 that '.E..x C y/jA// D '.E..x/ C .y/jA//. Thus, ' is additive on the positive part of F . Hence, ' admits an extension as a continuous linear functional on F . Evidently, ' is symmetric. In general, (a) follows from the above argument and the decomposition of a general symmetric functional into a linear combination of positive symmetric functionals as shown in Theorem 4.2.2. The proof of (b) is trivial. Proof of Theorem 4.4.1. (a) Let E be a symmetric function space. Let A, B 2 E.M, /C . It follows from Lemma 3.4.4 that .A C B/ C .A/ C .B/ C 21=2 .A C B/. If the functional ' is positive, then it follows from Lemma 4.4.3 that '..A C B// '..A/ C .B// '.21=2 .A C B// D '..A C B//. Thus, L.'/ is additive on E.M, /C and, therefore, it admits an extension as a linear functional on E.M, /. Evidently, this functional is symmetric. The assertion is proved for the case of a positive functional '. This is sufficient as, by Theorem 4.2.2, a general symmetric functional on E decomposes into a linear combination of positive symmetric functionals. If E is a symmetric sequence space then, due to Theorem 4.4.4, the argument is reduced to the consideration of the function space F . (b) The proof of (b) is identical to (a) since, due to Theorem 3.1.1, the Calkin correspondence is bijective in the atomless or atomic case.

120

Chapter 4 Symmetric Functionals

4.5 Figiel–Kalton Theorem Theorem 4.4.1 establishes that the Calkin correspondence provides a lift from symmetric functionals on symmetric function (or sequence) spaces to symmetric functionals on symmetric operators spaces, which is bijective in the atomless (or atomic) case. This important result is central to the theory of continuous traces, since we can use the classical Banach space theory of symmetric function spaces to answer questions concerning the existence of continuous traces. The Figiel–Kalton theorem is a fundamental result that can be used to show the existence of symmetric functionals on symmetric function spaces; we use it in subsequent sections. Let E be a symmetric function space either on the interval .0, 1/ or on the semi-axis. Define the sets DE :D Lin.fx 2 E : x D .x/g/ D f.a/  .b/, a, b 2 Eg, ZE :D Lin.fx1  x2 : 0 x1, x2 2 E, .x1 / D .x2 /g/. The set ZE is the common kernel of all linear functionals f : E ! C such that f .x1/ D f .x2 / if .x1 / D .x2 /. It is not quite the common kernel of all symmetric functionals since we have not demanded that f is continuous. The Figiel–Kalton theorem gives a description of ZE in terms of the Cesàro operator C : .L1 C L1 /.0, 1/ ! S.0, 1/, .C x/.t / D

1 t

Z

t

x.s/ds,

x 2 .L1 C L1 /.0, 1/.

0

Theorem 4.5.1 (Figiel–Kalton). Let E be a symmetric function space on the semiaxis and let x 2 DE . We have x 2 ZE if and Ronly if C x 2 E. The same assertion is 1 also valid for the interval .0, 1/ provided that 0 x.s/ds D 0. The theorem is proved below. Lemma 4.5.2 and Lemma 4.5.3 provide the proof of the “only if” part of Theorem 4.5.1.

Lemma 4.5.2. Let E be a symmetric function space. We have x 2 ZE H) C..xC /  .x // 2 E. Proof. Let x D Set

Pn

kD1 .xk

 yk / with xk , yk 2 EC and .xk / D .yk /, 1 k n.

z D xC C

n X kD1

yk D x C

n X kD1

xk .

121

Section 4.5 Figiel–Kalton Theorem

It follows from the definition of C and Lemma 3.3.3 that C.z/ C.xC/ C

n X

C.yk / D C..xC /  .x // C C.x / C

kD1

n X

C.xk /.

kD1

Using Lemma 3.3.5, we obtain that Z

t

..s, x / C

0

n X

Z

.nC1/t

.s, xk //ds 0

kD1

Z

t

.s, z/ds

.s, z/ds C nt.t , z/.

0

Therefore, C.z/ C.z/ C C..xC /  .x // C n.z/. It follows that C..x /  .xC // n.z/. Similarly, C..xC /  .x // n.z/ and we conclude that jC..xC /  .x //j n.z/ 2 E. Lemma 4.5.3. Let E be a symmetric function space. We have x 2 DE H) C..xC /  .x // 2 C x C E. Proof. It follows from the definition of DE that x D .a/  .b/ for some a, b 2 E. Observe that .a/  .a/ supp.xC /  x supp.xC / D xC . Set u :D .a/  xC  0. Clearly, .a/ D u C xC and .b/ D u C x . It follows from the definition of C and Lemma 3.3.3 that C.a/ C.u/ C C.xC / D C..xC /  .x // C C.u/ C C.x /. Using Lemma 3.3.5, we obtain that C.x / C C.u/ C.b/ C .b/. It follows that C x C..xC /  .x // C .b/. Similarly, C x  C..xC /  .x //  .a/, and the assertion follows.

122

Chapter 4 Symmetric Functionals

We now proceed with Lemma 4.5.5, which provides the proof of the “if” part of Theorem 4.5.1. We need the following deep theorem due to Kwapien [141]. We omit the proof of Theorem 4.5.4 as it falls beyond the scope of this book. R1 Theorem 4.5.4. For every x 2 L1 .0, 1/ such that 0 x.s/ds D 0, there exist positive functions y1 , y2 2 L1 .0, 1/ such that .y1 / D .y2 /, x D y1  y2 and ky1k1 , ky2 k1 6kxk1. Lemma 4.5.5. Let E be a symmetric function space on .0, 1/ and let x 2 DE . If C x 2 E, then x 2 ZE . The same assertion holds for R 1such an element x of a symmetric function space on the interval .0, 1/ provided that 0 x.s/ds D 0. Proof. Define a partition A :D f.2n, 2nC1 gn2Z and set x1 :D E.xjA/. If x D .a/ .b/ with a, b 2 E, then x1 D E..a/jA/  E..b/jA/. Clearly, E..a/jA/ 2 .a/ 2 E,

E..b/jA/ 2 .b/ 2 E

are decreasing functions. It follows that x1 2 DE . It is easy to see that jC x1  C xj 22..a/ C .b//. Since .a/, .b/, C x 2 E, it follows that C x1 2 E. Define a function z 2 E by setting z.t / :D .C x1/.2nC1 /,

t 2 .2n , 2nC1/.

Clearly, x1 D 2z  2z 2 ZE . Consider the function x  x1 on the interval .2n , 2nC1 /. By Theorem 4.5.4, there exist positive functions y1n , y2n supported on .2n , 2nC1/ such that .x  x1/ .2n ,2nC1 / D y1n  y2n and .y1n / D .y2n /,

ky1n k1 , ky2n k1 6k.x  x1/ .2n ,2nC1 / k1 .

Set y1 :D

X n2Z

y1n ,

y2 :D

X

y2n .

n2Z

It follows from the above inequality that y1 , y2 2 EC . Since x  x1 D y1  y2 and .y1 / D .y2 /, it follows that x  x1 2 ZE . Since we already established that x1 2 ZE , it follows that x 2 ZE . Proof of Theorem 4.5.1. If x 2 ZE \ DE , then it follows from Lemma 4.5.2 that C..xC /  .x // 2 E. By Lemma 4.5.3, we have that C x 2 E.

Section 4.6 Existence of Symmetric Functionals

123

Conversely, if x 2 DE (if E is symmetric function space on the interval .0, 1/, then R1 we require also that 0 x.s/ds D 0) and C x 2 E, then it follows from Lemma 4.5.5 that x 2 ZE .

4.6 Existence of Symmetric Functionals In this section, we present results concerning the existence of (singular) symmetric functionals on symmetric function spaces. In Section 4.9 we spell out the consequences of the existence results shown in this section for the existence of symmetric functionals on symmetric operator spaces. Theorem 4.6.1. Let E be a symmetric function space and let 0 x 2 E. (a) If E D E.0, 1/, then there exists a positive symmetric functional ' 2 E  such that 1 '.x/ D lim km .x/kE . m!1 m (b) If E D E.0, 1/  L1.0, 1/, then there exists a positive singular symmetric functional ' 2 E  such that '.x/ D lim

m!1

1 k.m .x// .0,1/ kE . m

(c) If E D E.0, 1/, then there exists a positive singular symmetric functional ' 2 E  such that 1 '.x/ D lim km .x/kE . m!1 m Theorem 4.6.1 is proved below. The next result is that every symmetric functional on a symmetric function space decomposes into normal and singular symmetric functionals. Theorem 4.6.2. Let E be a symmetric function space and let ' 2 E  be a symmetric functional. (a) If E D E.0, 1/ 6 L1.0, 1/, then ' is singular. (b) If E D E.0, 1/  L1 .0, 1/, or E D E.0, 1/, then there exists a singular symmetric functional 's 2 E  such that ' D ˛m C 's for a scalar ˛. Here m denotes the Lebesgue integral. To prove the theorems we need the following variant of the Hahn–Banach theorem for partially ordered spaces, and the subsequent lemmas.

124

Chapter 4 Symmetric Functionals

Lemma 4.6.3. Let E be a partially ordered linear space and let p : E ! R be a convex and monotone functional (that is, p.x/ p.y/ provided that x y and x, y 2 E). For every x0 2 E, there exists a positive linear functional ' : E ! R such that ' p and '.x0/ D p.x0 /. Proof. The existence of a linear functional ' p such that '.x0/ p.x0/ follows from the Hahn–Banach theorem. We only have to prove that '  0. If z  0, then, by assumption, p.x0/  p.x0  z/. Therefore, '.z/ D '.x0 /  '.x0  z/ D p.x0/  '.x0  z/  p.x0/  p.x0  z/  0. Recall that the sets DE and ZE were introduced at the beginning of Section 4.5. Lemma 4.6.4. Let E be a symmetric function space on the semi-axis. If the mapping  : E ! E is defined by setting .x/ :D .xC /  .x /, then (a)  : E ! DE and  : ZE ! DE \ ZE . For every x 2 E, we have .x/ 2 x C ZE . (b) for every x, y 2 E, we have .x C y/ 2 .x/ C .y/ C DE \ ZE . Proof. We derive (b) from (a). (a) The first assertion is trivial. For every x 2 E, it follows from the definition of ZE that .xC /  xC 2 ZE and .x /  x 2 ZE . Hence, .x/  x D ..xC /  xC/  ..x /  x / 2 ZE . This proves the third assertion. The second assertion follows by observing that .x/, x 2 E belongs to DE and that .x/ 2 ZE CZE D ZE for every x 2 ZE . (b) It follows from (a) that .x C y/ 2 x C y C ZE ,

.x/ 2 x C ZE ,

.y/ 2 y C ZE

and, therefore, .x C y/ 2 .x/ C .y/ C ZE . The result is shown since  : E ! DE .

125

Section 4.6 Existence of Symmetric Functionals

Lemma 4.6.5. Let E be a symmetric function space on the semi-axis. Let p : DE ! R be a convex and monotone functional. If p D 0 on ZE \ DE , then (a) p D p ı  on DE , where  is the mapping in Lemma 4.6.4. (b) p extends to a convex monotone functional p : E ! R by setting p D p ı . (c) p D 0 on ZE . Proof. It is stated in Lemma 4.6.4 (a) that .x/  x 2 ZE for every x 2 E. From the same lemma we also have that .x/ 2 DE . Thus, .x/  x 2 DE for every x 2 DE and we conclude that .x/ D x C z for some z 2 DE \ ZE . It follows from the assumptions that p..x// p.x/ C p.z/ D p.x/,

p.x/ p..x// C p.z/ D p..x//.

This proves (a). It follows from Lemma 4.6.4 that .x C y/ D .x/ C .y/ C z with z 2 DE \ ZE for every x, y 2 E. It follows from the assumptions that p..x C y// p..x// C p..y// C p.z/ D p..x// C p..y//. This proves the convexity of the extension p ı . Since both the functional p : DE ! R and the mapping  : E ! E are monotone, it follows that the extension p ı  is also monotone. This proves (b). If x 2 ZE , then .x/ 2 DE \ ZE . Thus, p.x/ D p..x// D 0, which proves (c).

Lemma 4.6.6. If 0 x 2 .L1 C L1 /.0, 1/, then   Z b Z b  Z b=m 1 1 1  m C x .s/ds x.s/ds x.s/ds log.m/ a m a a=m provided that ma b. In particular,

  1 1 1  m C x C x m m x C log.m/ m 1

provided that x D .x/. Proof. Clearly, 1 log.m/

   Z t 1 1 x.s/ds, 1  m C x .t / D m t log.m/ t =m

t > 0.

126

Chapter 4 Symmetric Functionals

Therefore, Z a

b



  Z bZ t 1 dt 1  m C x .s/ds D x.s/ds m t a t =m Z b Z minfms,bg dt D x.s/ds t a=m maxfa,sg   Z b minfms, bg x.s/ log D ds. maxfa, sg a=m

The integrand does not exceed x.s/ log.m/ and the second inequality follows immediately. The integrand is positive and is equal to x.s/ log.m/ for s 2 .a, b=m/. The first inequality follows. Lemma 4.6.7. Let E be a symmetric function space on the semi-axis. The functional p : DE ! R defined by setting    1 1 1  m C x , x 2 DE , p.x/ :D lim sup m!1 log.m/ m C E satisfies the assumptions of Lemma 4.6.5. Also, for every x 2 DE , we have p.x/ kxkE . Proof. If x 2 DE , then x D .a/  .b/ for some a, b 2 E. Set z D .a/ C .b/ 2 E. It follows from Lemma 4.6.6 and Corollary 3.4.3 that      1  1 m C x 1  1 m C z log.m/  kzkE . m m C E E Thus, p.x/ is finite for all x 2 DE . Observe that the mappings, m  1,    1 x! 1  m C x , m C are convex and monotone. So are the mappings    1 x ! 1  m C x , m C E

x 2 E,

x 2 E.

Therefore, p : DE ! R is a convex and monotone functional. If x 2 ZE \ DE , then, by Theorem 4.5.1, we have that C x 2 E. Therefore,    1  1 m C x kC xkE C 1 km C xkE 2kC xkE . m m C E Letting m ! 1, we obtain p.x/ D 0.

Section 4.6 Existence of Symmetric Functionals

127

The following lemma shows that every symmetric functional on E.0, 1/ 6 L1.0, 1/ is necessarily a singular symmetric functional. Lemma 4.6.8. If the symmetric function space E on the semi-axis is such that E 6 L1.0, 1/, then every symmetric functional ' on E is singular. Proof. Let x 2 E be bounded and finitely supported. We have .x/ kxk1 .0,n/ for some n 2 N. Therefore, ˇ  ˇ ˇ ˇ 1 1 m .x/ ˇˇ k'kE   km .x/kE j'.x/j D ˇˇ' m m 1 k'kE  kxk1  k .0,nm/ kE . m Letting m ! 1 and taking into account that E 6 L1.0, 1/, we obtain that the righthand side tends to 0. Therefore, '.x/ D 0 for every bounded and finitely supported x. By continuity, '.x/ D 0 for every x 2 .L1 \ L1 /.0, 1/. Hence, the functional ' is singular. The case E  L1 requires a more detailed treatment. Lemma 4.6.9. Let E be a symmetric (respectively, fully symmetric) function space either on the interval .0, 1/ or on the semi-axis. Let f'i gi 2I 2 E  be a net and let ' 2 E  be such that 'i ! ' in the weak topology. (a) If every 'i is symmetric, then ' is symmetric. (b) If every 'i is fully symmetric, then ' is fully symmetric. Proof. Let each 'i be a symmetric functional on E. If 0 x1, x2 2 E are such that .x1 / D .x2 /, then '.x1 / D lim 'i .x1/ D lim 'i .x2 / D '.x2 /. i 2I

i 2I

Hence, ' is a symmetric functional on E. Let each 'i be a fully symmetric functional on E. If 0 x1, x2 2 E are such that .x2 /  .x1 /, then '.x2 / D lim 'i .x2 / lim 'i .x1 / D '.x1/. i 2I

i 2I

Hence, ' is a fully symmetric functional on E. Lemma 4.6.10. Let E be a symmetric (respectively, fully symmetric) function space and let ' be a positive symmetric (respectively, fully symmetric) functional on E. The

128

Chapter 4 Symmetric Functionals

formula 'sing .x/ :D lim '..x/ .0,1=n/ /, n!1

0 x2E

defines a positive singular symmetric (respectively, fully symmetric) linear functional on E. Proof. If x, y 2 E are positive functions, then it follows from Lemma 3.4.4 that .x C y/ .0,1=n/ C ..x/ C .y// .0,1=n/ C 21=2 ..x C y/ .0,2=n/ /. Taking the limit as n ! 1, we derive from Lemma 4.4.3 that 'sing ..x C y// D 'sing ..x/ C .y//. Since ' is symmetric, it follows that 'sing .x C y/ D 'sing ..x C y// D 'sing ..x/ C .y// D 'sing .x/ C 'sing .y/. Hence, 'sing is an additive functional on EC . Therefore, it extends to a linear functional on E. Clearly, 'sing is symmetric. If ' is fully symmetric, then the fact that 'sing is fully symmetric is trivial. Lemma 4.6.11. Let E be a symmetric function space on the semi-axis and let ' be a positive symmetric functional on E. If E  L1.0, 1/ and if 'sing is the functional defined in Lemma 4.6.10, then '  'sing is a scalar multiple of the Lebesgue integral. Proof. Fix a positive function z 2 .L1 \ L1 /.0, 1/. For  > 0, select a positive step function u 2 .L1 \ L1 /.0, 1/ such that kz  ukL1\L1 . Since u is a step function, it follows that '.u/ D kuk1'. .0,1/ /. Since ' is a continuous function, it follows that j'.z/  kzk1 '. .0,1/ /j j'.z  u/j C ku  zk1 '. .0,1/ / k'kE   ku  zkE C ku  zk1  k'kE  2k'kE   ku  zkL1\L1 2k'kE  . Since  is arbitrarily small, it follows that '.z/ D kzk1'. .0,1/ /. It now follows from the definition of 'sing that, for every positive x 2 E, Z 1 .'  'sing /.x/ D lim '..x/ .1=n,1/ / D '. .0,1/ /  .s, x/ds. n!1

The assertion follows from the linearity of '  'sing . Now, we prove Theorems 4.6.1 and 4.6.2.

0

129

Section 4.6 Existence of Symmetric Functionals

Proof of Theorem 4.6.1 (a). Without loss of generality, x D .x/. Let p be the convex monotone functional constructed in Lemma 4.6.7. It follows from Lemma 4.6.3 that there exists a positive linear functional ' on E such that ' p and '.x/ D p.x/. Since p.z/ D 0 for every z 2 ZE , it follows that '.z/ D 0 for every z 2 ZE . Therefore, ' is a symmetric functional. For every z D .z/ 2 E, it follows from Lemma 4.6.6 and Corollary 3.4.3 that p.z/ kzkE . Hence, '.z/ p.z/ kzkE for every z D .z/ 2 E. It follows that k'kE  1. Therefore,   1 1 m x km xkE . '.x/ D ' m m Taking the limit m ! 1, we obtain '.x/ lim

m!1

1 km .x/kE . m

On the other hand, it follows from Lemma 4.6.6 that   1 1 1 1  m C x. m m x C log.m/ m Therefore,

  1 1 1  lim 1  m C x km .x/kE . p.x/ D lim sup m!1 log.m/ m!1 m m E

The assertion follows immediately. Proof of Theorem 4.6.1 (b). Fix j  1 and apply Theorem 4.6.1 (a) to the function .x/ .0,1=j / . It follows that there exists a symmetric linear functional 'j such that k'j kE  1 and 1 1 km ..x/ .0,1=j / /kE  lim km ..x// .0,1/ kE . m!1 m m!1 m

'j ..x/ .0,1=j / / D lim

Since the unit ball in E  is weak compact (the Banach–Alaoglu theorem), there exists a convergent subnet i D 'F .i /, i 2 I, of the sequence 'j , j 2 N. Let i ! ' in the weak topology. It follows from Lemma 4.6.9 that ' is a symmetric functional. By the definition of a subnet (see [188, Section IV.2]), for every fixed j 2 N, there exists ij 2 I such that F .i/ > j for every i > ij . Thus, for every ij < i 2 I, we have i ..x/ .0,1=j / /

The subnet

i , ij

 'F .i / ..x/ .0,1=F .i // /  lim

m!1

1 km ..x// .0,1/ kE . m

< i 2 I converges to the same limit '. Therefore, '..x/ .0,1=j / /  lim

m!1

1 km ..x// .0,1/ kE . m

130

Chapter 4 Symmetric Functionals

Now, taking the limit as j ! 1 and using Lemma 4.6.10, we obtain the inequality 1 km ..x// .0,1/ kE , m!1 m

'sing .x/  lim

where 'sing is the singular symmetric functional defined in Lemma 4.6.10. The opposite inequality can be obtained by the same argument as in the proof of Theorem 4.6.1 (a). Proof of Theorem 4.6.1 (c). Let F be the symmetric function space (on the semi-axis) consisting of all measurable functions x on .0, 1/ that are finite for the norm given by the formula kxkF :D k.x/ .0,1/ kE C kxk1. Clearly, F  L1.0, 1/. Applying Theorem 4.6.1 (b), we obtain a singular symmetric functional ' on F such that '.x/ D lim

m!1

1 1 km ..x// .0,1/ kF D lim km ..x//kE . m!1 m m

Proof of Theorem 4.6.2 (a). Lemma 4.6.8 provides the result for positive symmetric functionals. We then use the Jordan decomposition in Theorem 4.2.2. Proof of Theorem 4.6.2 (b). Lemma 4.6.11 provides the result for positive symmetric functionals. We then use the Jordan decomposition in Theorem 4.2.2.

4.7 Existence of Fully Symmetric Functionals In this section we consider the existence of fully symmetric functionals on fully symmetric function spaces. In Theorem 4.7.1 the reader will notice identical existence criteria to Theorem 4.6.1 in the last section. Again, the consequences of the results of this section for the existence of fully symmetric functionals on fully symmetric operator spaces are discussed in Section 4.9. Theorem 4.7.1. Let E be a fully symmetric function space and let 0 x 2 E. (a) If E D E.0, 1/, then there exists a fully symmetric functional ' 2 E  such that '.x/ D lim

m!1

1 km .x/kE . m

(b) If E D E.0, 1/  L1.0, 1/, then there exists a singular fully symmetric functional ' 2 E  such that '.x/ D lim

m!1

1 k.m .x// .0,1/ kE . m

131

Section 4.7 Existence of Fully Symmetric Functionals

(c) If E D E.0, 1/, then there exists a singular fully symmetric functional ' 2 E  such that 1 km .x/kE . '.x/ D lim m!1 m The following lemmas, in combination with some of those in the last section, prove the theorem. Lemma 4.7.2. If x D .x/ and y D .y/ are such that y  x, then     1 1 1  m Cy  1  m C x. m m Proof. Arguing as in Lemma 4.6.6, we have that     Z t  Z t 1 minfms, t g ds. 1  m C z .s/ds D z.s/ log m s 0 0 For a fixed t > 0, the mapping  s ! log

 minfms, t g , s

s>0

is decreasing. It now follows from [139, Equality 2.36] that     Z t  Z t  1 1 1  m Cy .s/ds 1  m C x .s/ds m m 0 0 for every t > 0. Since the functions     1 1 1  m Cy, 1  m C x m m are both decreasing, the assertion follows. Lemma 4.7.3. Let E be a fully symmetric function space on the semi-axis and let x D .x/ 2 E. If p is the convex functional defined in Lemma 4.6.7 and if z 2 DE is such that C x C z, then p.x/ p.z/. Proof. By assumption, z D .a/  .b/ with a, b 2 E. Since C x C z, it follows that C.x C .b// C.a/ or, equivalently, x C .b/  .a/. It follows from Lemma 4.7.2 that, for every t > 0,     Z t  Z t  1 1 1  m C.x C .b// .s/ds 1  m C.a/ .s/ds m m 0 0

132

Chapter 4 Symmetric Functionals

and, therefore,     Z t  Z t  1 1 1  m C x .s/ds 1  m C z .s/ds m m 0 0   Z t  1 1  m C z .s/ds m 0 C    Z t   1 ds.  s, 1  m C z m 0 C The assertion now follows from the definition of the functional p (given in Lemma 4.6.7). Lemma 4.7.4. Let E be a fully symmetric function space on the semi-axis. Let p be the convex functional defined in Lemma 4.6.7. The functional q.x/ :D inffp.z/ : z 2 DE , C x C zg,

x 2 DE

satisfies the assumptions of Lemma 4.6.5. Proof. Let x1, x2 2 DE . Fix  > 0 and select z1 , z2 2 DE such that C xi C zi and p.zi / q.xi / C  for i D 1, 2. Thus, C.x1 C x2/ C.z1 C z2 / and, by convexity of the functional p, q.x1 C x2 / p.z1 C z2 / p.z1 / C p.z2 / q.x1/ C q.x2 / C 2. Since  is arbitrarily small, the convexity of the functional q follows. Let x1 , x2 2 DE be such that x1 x2. Fix  > 0 and select z 2 DE such that C x2 C z and p.z/ q.x2 / C . Thus, C x1 C x2 C z and q.x1 / p.z/ q.x2/ C . Since  is arbitrarily small, it follows that the functional q is monotone. For x 2 ZE \ DE , we have 0 q.x/ p.x/ D 0 and, therefore, q.x/ D 0. Proof of Theorem 4.7.1 (a). Without loss of generality, x D .x/. Let q be the convex monotone functional constructed in Lemma 4.7.4. It follows from Lemma 4.6.3 that there exists a positive linear functional ' on E such that ' q and '.x/ D q.x/. By the construction of the functional q, we have q p and, therefore ' p. Since p.z/ D 0 for every z 2 ZE , it follows that '.z/ D 0 for every z 2 ZE . Therefore, ' is a symmetric functional. Let x1, x2 2 E be positive elements such that x1  x2 . Therefore, z D .x1 /  .x2 / 2 DE and C z 0. It follows from the workings above that '.z/ q.z/ p.0/ D 0. Hence, ' is a fully symmetric functional. For every z D .z/ 2 E, it follows from Lemma 4.6.6 and Corollary 3.4.3 that p.z/ kzkE . Hence, '.z/ q.z/ p.z/ kzkE for every z D .z/ 2 E. It

Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different

133

follows that k'kE  1. Therefore,   1 1 m x km xkE . '.x/ D ' m m Taking the limit m ! 1, we obtain '.x/ lim

m!1

1 km .x/kE . m

On the other hand, it follows from the fact that x D .x/ and Lemma 4.7.3 that q.x/ D p.x/. By Lemma 4.6.6, we have   1 1 1  m C x. m1 m x C log.m/ m Therefore,

  1 1 1 km .x/kE .  lim '.x/ D q.x/ D p.x/ D lim sup 1 m C x m!1 log.m/ m!1 m m E The assertion follows immediately. The proofs of Theorem 4.7.1 (b),(c) are very similar to that of Theorem 4.6.1 and are, therefore, omitted. The only difference is that the reference to Theorem 4.6.1 (a) has to be replaced with the reference to Theorem 4.7.1 (a).

4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different The last section gave criteria for the existence of a fully symmetric functional on a fully symmetric function space. Are there symmetric functionals that are not fully symmetric on a fully symmetric function space? In Theorem 4.8.1 we demonstrate that the sets of symmetric and fully symmetric functionals on a given fully symmetric space E equipped with a Fatou norm are distinct provided that a nontrivial singular symmetric functional exists. Theorem 4.8.1. Let E be a fully symmetric function space equipped with a Fatou norm and let 0 x 2 E. Let '.y/ '.x/ for every 0 y  x and for every positive symmetric functional ' 2 E  . (a) If E D E.0, 1/ 6 L1.0, 1/, then m1 m .x/ ! 0 in the norm topology of E. (b) If E D E.0, 1/  L1 .0, 1/, then m1.m .x// .0,1/ ! 0 in the norm topology of E. (c) If E D E.0, 1/, then m1m .x/ ! 0 in the norm topology of E.

134

Chapter 4 Symmetric Functionals

The theorem says, in combination with Theorem 4.7.1 and Theorem 4.2.2, that if every symmetric functional on E is a fully symmetric functional, then all symmetric functionals are trivial. By contradiction then, if a nontrivial fully symmetric functional exists then there must be a distinct symmetric functional. The following lemma is the key technical step in separating fully symmetric functionals from symmetric ones. Lemma 4.8.2. Let E be a fully symmetric function space. If x, y 2 E are such that '.y/ '.x/ for every positive symmetric functional ' 2 E  , then Z

Z

b

mb

.s, y/ds

ma

a

..s, x/ C um .s// ds,

ma < b,

where 0 um ! 0 in E. Proof. Without loss of generality, x D .x/ and y D .y/. Let p be a convex positive functional constructed in Lemma 4.6.7. By Lemma 4.6.3, there exists a positive functional ' 2 E  such that ' p and '.y  x/ D p.y  x/. By construction of p, we have p.z/ D 0 for every z 2 ZE and, therefore, '.z/ D 0 for every z 2 ZE . Therefore, ' is a positive symmetric linear functional on E. By assumption, '.y  x/ 0. Therefore, p.y  x/ D '.y  x/ 0. Since p takes only positive values, it follows that p.y  x/ D 0. Hence, by the definition of p, we have    1 1 um :D 1  m C.y  x/ !0 log.m/ m C in E. It follows from the definition of the convex functional p (see Lemma 4.6.7) that     1 1 1 1 (4.7) 1  m Cy 1  m C x C um . log.m/ m log.m/ m By Lemma 4.6.6, we have Z

b

1 y.s/ds log.m/ ma

Z

mb ma

   1 1  m Cy .s/ds. m

It now follows from (4.7) and Lemma 4.6.6 that Z

b

Z

mb

Z

ma mb

y.s/ds

ma



a

 um C

   1 1 1  m C x .s/ds log.m/ m

.x C um /.s/ds.

135

Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different

For each sequence  and  > 0, we define the sequence   by setting ( .n/, .n/     .n/ :D 1, .n/ < . Below, we equip the set .RC [ f1g/Z with componentwise partial ordering. Proposition 4.8.3. Let the mappings Ãm : .RC [ f1g/Z ! RC , m  1, be quasidecreasing (that is, there exists a constant const such that  0  implies Ãm ./ const  Ãm . 0 /). Assuming that (a) for every m  1, we have Ãm .n / ! Ãm ./ when n #  (b) there exists 0 2 RZC such that Ãm ./ D Ãm . ^ 0 / for all  2 .RC [ f1g/Z, m1 (c) for every  2 .RC [ f1g/Z, we have Ãm . m / ! 0 as m ! 1 (d) for every  2 .RC [ f1g/Z, the sequence Ãm ./, m  1, decreases we have Ãm .m/ ! 0 as m ! 1. Proof. By the assumptions, we have that     m m (a) (b) (b) Ãm D Ãm ^ 0 ! Ãm .m ^ 0 / D Ãm .m/,

.r ,r /

.r ,r / For every m  1, select rm sufficiently large so that   m 1 Ãm  Ãm .m/.

.rm ,rm / 2

r ! 1.

(4.8)

Without loss of generality, rm " 1 as m ! 1. Now define the element  2 RZC by setting ^ m  :D .

.rm ,rm / m1

Clearly, .n/ > m for jnj  rm and, therefore .n/ ! 1 as jnj ! 1. Fix  > 0 and select  such that à .  / <  (this can be done due to (c)). The set fn : .n/ < g is finite. For an arbitrary m such that m > maxf, max 0 .n/g, .n/ 0.

(4.13)

0

This proves that Ãm is quasi-decreasing. If k # , then E.xjB k / converges to E.xjB / almost everywhere. Since the norm on E is a Fatou norm, it follows that Ãm is order-continuous on the right. Hence, the condition (b) of Proposition 4.8.3 holds. Setting 0 .n/ :D a.3n C 1/=a.3n/ for every n 2 Z, we satisfy the condition (b) of Proposition 4.8.3. By Proposition 4.8.4, the condition (c) of Proposition 4.8.3 also holds. The assertion now follows from Proposition 4.8.3. Lemma 4.8.6. If x D .x/ 2 L1 C L1 and if Ci , 1 i k, are discrete sets, then E.xj [kiD1 Ci / 

k X

E.xjCi /.

i D1

Proof. It is sufficient to verify the inequality Z

t

E.xj

0

[kiD1

Ci /.s/ds

k Z X

t

E.xjCi /.s/ds

i D1 0

only at the nodes of E.xj[kiD1 Ci /, that is at the nodes of E.xjCi / for every i. However, if t 2 Ci for some i, then Z t Z t E.xj [kiD1 Ci /.s/ds D X.t / D E.xjCi /.s/ds 0

0

and the result is shown. Set Am :D fma.n/ : where n 2 Z is such that m2 a.n/ < a.n C 1/g.

Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different

139

Lemma 4.8.7. Let E be a fully symmetric function space equipped with a Fatou norm. If x D .x/ 2 E is such that '.y/ '.x/ for every positive symmetric functional ' on E and every 0 y  x, then m1m E.xjAm / ! 0 in the norm topology of E. Proof. Let Cm :D fma.3n C 1/ : where n 2 Z is such that m2 a.3n C 1/ < a.3n C 2/g, Dm :D fma.3n C 2/ : where n 2 Z is such that m2 a.3n C 2/ < a.3n C 3/g. It is clear that Bm [ Cm [ Dm D Am . Therefore, by Lemma 4.8.6, we have E.xjAm /  E.xjBm / C E.xjCm / C E.xjDm /.

(4.14)

By Proposition 4.8.5, we have m1 m E.xjBm / ! 0 as m ! 1. Similarly, applying Proposition 4.8.5 to the functions 2x=3 and 4x=9, we obtain that m1 m E.xjCm / ! 0 and m1 m E.xjDm / ! 0 as m ! 1. The assertion follows immediately. Lemma 4.8.8. Let x D .x/ 2 .L1 C L1 /.0, 1/. If x … L1 .0, 1/, then, for every t > 0 and every m 2 N, we have Z 4 2 3 mt 4 X.t / X.m t / C E.xjAm /.s/ds, (4.15) 3 2 0 Rt where X.t / :D 0 x.s/ds. Proof. For a given t > 0, there exists n 2 Z such that t 2 Œa.n/, a.n C 1/. We consider all the cases for the relationships amongst a.n/, a.n C 1/ and a.n C 2/. If a.n C 1/ > m2a.n/, then m4 t  ma.n/ and, therefore, Z m4 t Z ma.n/ 2 E.xjAm /.s/ds  E.xjAm /.s/ds D X.ma.n//  X.t /. 3 0 0 If a.n C 1/ m2 a.n/ and a.n C 2/ > m2a.n C 1/, then m4 t  ma.n C 1/ and, therefore, Z m4 t Z ma.nC1/ E.xjAm /.s/ds  E.xjAm /.s/ds D X.ma.n C 1//  X.t /. 0

0

If a.n C 2/ m2 a.n C 1/ and a.n C 1/ m2a.n/, then X.m4t /  X.a.n C 2// D and the assertion follows.

3 3 X.a.n C 1//  X.t / 2 2

140

Chapter 4 Symmetric Functionals

The situation in the case where x 2 L1 is slightly more complicated. Lemma 4.8.9. If x D .x/ 2 L1 .0, 1/ or x 2 L1.0, 1/, then there exists constant C such that for every t > 0 2 3 X.t / X.m4 t / C 3 2 where X.t / :D

Rt 0

Z

m4 t

Z

m4 t

E.xjAm /.s/ds C C

0

Œ0,1 .s/ds,

(4.16)

0

x.s/ds.

Proof. Consider first the case of the semi-axis. Fix n0 such that X.a.n0 // 4=9X.1/. For a given t 2 .0, a.n0 //, there exists n 2 Z such that n < n0 and such that t 2 .a.n/, a.n C 1//. Then, the same argument in Lemma 4.8.8 applies. For every t  a.n0 / we have X.t /

X.1/ X.1/ minfm4t , 1g D minfa.n0 /, 1g minfa.n0 /, 1g

Z

m4 t

Œ0,1 .s/ds. 0

Setting C :D X.1/= minfa.n0 /, 1g, we obtain the assertion. The same argument applies in the case of the interval .0, 1/ by replacing X.1/ with X.1/. Proof of Theorem 4.8.1 (a). Without loss of generality, x D .x/. If x … L1 , then by Lemma 4.8.8, Z

t =m4 0

2 x.s/ds 3

Z

t 0

3 x.s/ds C 2

Z

t

E.xjAm /.s/ds,

8t > 0

0

or, equivalently, 1 2 3  4 x  x C E.xjAm /. m4 m 3 2 Applying m1 m to both sides of the last display, we obtain 1 21 31 m5 x  m x C m E.xjAm /. 5 m 3m 2m We now apply the norm k  kE to both sides of the last display and let m ! 1. It follows from Lemma 4.8.7 that lim

m!1

This proves the assertion.

1 1 2 km xkE lim km xkE . m!1 m 3 m

141

Section 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different

If x 2 L1 and C are as in Lemma 4.8.9, then it follows from Lemma 4.8.9 that Z

t =m4

x.s/ds

0

2 3

Z

t

x.s/ds C

0

3 2

Z

t

Z E.xjAm /.s/ds CC

0

t

Œ0,1 .s/ds,

8t > 0

0

or, equivalently, 2 3 1 m4 x  x C E.xjAm / C C .0,1/. 4 m 3 2 Applying m1m to both sides of the last display, we obtain 1 21 31 1 m x C m E.xjAm / C C m .0,1/ .  5 x  m5 m 3m 2m m We apply the norm k  kE to both sides of the last display and let m ! 1. For every symmetric space E on the interval .0, 1/ and for every symmetric space E on the semi-axis such that E 6 L1 .0, 1/ we have m1km .0,1/ kE ! 0. It follows from Lemma 4.8.7 that lim

m!1

2 1 1 km xkE lim km xkE m!1 m 3 m

and the assertion follows. Proof of Theorem 4.8.1 (b). Consider the Banach space .E C L1 /.0, 1/ equipped with the norm given by the formula kxkECL1 :D k.x/ .0,1/ kE . Observe that E C L1 is a fully symmetric space with a Fatou norm. Suppose that the assertion of Theorem 4.8.1 (b) fails. That is, lim

m!1

1 km .x/kECL1 > 0. m

Applying Theorem 4.8.1 (a) to the space ECL1 , we find that there exists 0 y  x and a positive symmetric functional ' 2 .E C L1 / such that '.y/ > '.x/. Obviously, y 2 E and ' is a symmetric functional on E. Thus, we obtained a contradiction. This proves the assertion of Theorem 4.8.1 (b). The proof of Theorem 4.8.1 (c) is identical to that of Theorem 4.8.1 (a) and is, therefore, omitted.

142

Chapter 4 Symmetric Functionals

4.9 Symmetric Functionals on Symmetric Operator Spaces Sections 4.6–4.8 have provided results concerning the existence of symmetric functionals, fully symmetric functionals, and the distinction between the two, for symmetric and fully symmetric function spaces. We can use the lifting established in Section 4.4 to transfer these results to statements about the existence of symmetric functionals on symmetric operator spaces. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace . Theorem 4.9.1 below establishes criteria for when a nontrivial symmetric functional exists on a symmetric operator space E.M, / associated to a symmetric function space E. Theorem 4.9.2, which imposes the condition that M be an atomless or atomic semifinite von Neumann algebra, is a stronger result than Theorem 4.9.1. Using the Calkin correspondence we can associate to every symmetric operator space a symmetric function space, and bijectively lift the symmetrical functionals on the symmetric function space to symmetric functionals on the symmetric operator space. The assertion of Theorem 4.1.3 in the introduction to this chapter follows from Theorem 4.9.2 by setting M D L.H / and recalling, from Section 2.7, that the set of continuous traces on a symmetrically normed ideal of compact operators is identical to the set of symmetric functionals on that ideal. Theorem 4.9.1. Let E be a symmetric function (respectively, sequence) space with corresponding symmetric operator space E.M, / :D fA 2 S.M, / : .A/ 2 Eg , kAkE :D k.A/kE . (respectively E.M, / :D fA 2 S.M, / : E..A/jA/ 2 Eg , kAkE :D kE..A/jA/kE . / Consider the following conditions. (a) There exist nontrivial positive singular symmetric functionals on E.M, /. (b) There exist nontrivial singular fully symmetric functionals on E.M, /. (c) There exist positive symmetric functionals on E.M, / that are not fully symmetric functionals. (d) If .1/ D 1 and if E.M, / 6 L1 .M, /, then there exists an operator A 2 E.M, / such that 1 ˚m (4.17) lim kA kE > 0. m!1 m

143

Section 4.9 Symmetric Functionals on Symmetric Operator Spaces

If .1/ D 1 and if E.M, /  L1 .M, /, then there exists an operator A 2 E.M, / such that 1 ˚m lim kA kE CL1 > 0. (4.18) m!1 m If .1/ D 1, then there exists an operator A 2 E.M, / such that lim

m!1

1 ˚m kA k.E CL1 /.M˝l1 ,˝ 0 / > 0. m

(4.19)

Here,  0 is the semifinite trace on l1 given by the sum, as in Example 2.2.3 (a). (i)

The condition (d) implies (a).

(ii) The condition (d) implies the equivalent conditions (a), (b) if E is fully symmetric. (iii) The condition (d) implies the equivalent conditions (a)–(c) if E is fully symmetric and equipped with a Fatou norm. The proof of Theorem 4.9.1 is contained in the proof of the next theorem since none of the analogous implications (that is, in one direction) in Theorem 4.9.2 require the condition that the von Neumann algebra be atomless or atomic. In the atomic case we assume that the trace of every atom is 1. Theorem 4.9.2. Let M be an atomless or atomic von Neumann algebra equipped with a fixed faithful normal semifinite trace . Let E.M, / be a symmetric operator space. Consider the following conditions. (a) There exist nontrivial positive singular symmetric functionals on E.M, /. (b) There exist nontrivial singular fully symmetric functionals on E.M, /. (c) There exist positive symmetric but not fully symmetric functionals on E.M, /. (d) If .1/ D 1 and if E.M, / 6 L1 .M, /, then there exists an operator A 2 E.M, / such that 1 ˚m lim (4.20) kA kE > 0. m!1 m If .1/ D 1 and if E.M, /  L1 .M, /, then there exists an operator A 2 E.M, / such that 1 ˚m lim kA kE CL1 > 0. (4.21) m!1 m If .1/ D 1, then there exists an operator A 2 E.M, / such that 1 ˚m kA k.E CL1 /.M˝l1 ,˝ 0 / > 0. m!1 m lim

(4.22)

Here,  0 is the semifinite trace on l1 given by the sum, as in Example 2.2.3 (a).

144 (i)

Chapter 4 Symmetric Functionals

The conditions (a) and (d) are equivalent.

(ii) The conditions (a), (b) and (d) are equivalent if E.M, / is fully symmetric. (iii) The conditions (a)–(d) are equivalent if E.M, / is fully symmetric and equipped with a Fatou norm. Proof. The implications (b) ) (a) and (c) ) (a) are trivial. (a) ) (d) Let E.M, / be a symmetric operator space with a nontrivial singular symmetric functional '. Let A 2 E.M, / be an operator such that '.A/ ¤ 0. Without loss of generality, A  0. Let .1/ D 1 and let E.M, / 6 L1 .M, /. We have j'.A/j D

1 1 j'.A˚m /j k'kE   kA˚m kE . m m

Taking the limit as m ! 1, we obtain the required inequality (4.20). Let .1/ D 1 and let E.M, /  L1 .M, /. Note that this assumption prevents M from being atomic (indeed, otherwise E.M, / D L1 .M, / does not support any singular symmetric functional). Hence, M is atomless. Since ' is a singular functional (that is, ' vanishes on .L1 \ L1 /.M, /) and since     1 , A , 1 2 .L1 \ L1 /.M, /, 8m 2 ZC , A  AEA  m it follows that ˇ  ˇ   ˇ  ˚m ˇ     ˇ ˇ ˇ ˇ 1 1 1 ˇ , A , 1 ˇˇ D ˇˇ' AEA  ,A ,1 j'.A/j D ˇˇ' AEA  ˇ m m m     ˚m 1 1  ,A ,1 k'kE  AEA  m m E k'kE  

1 ˚m kA kE CL1 . m

Taking the limit as m ! 1, we obtain the required inequality (4.21). If .1/ D 1, then the proof of (4.22) is identical to that of (4.21) in the above paragraph (and is, therefore, omitted). (d) ) (a) Firstly, we assume that the algebra M is -finite. Without loss of generality, .1/ D 1. Let E.M, / be a symmetric operator space and let E be the corresponding symmetric function space. By the assumptions, there exists an element x D .A/ 2 E such that m1m x 6! 0 in E. By Theorem 4.6.1 (c), there exists a positive singular symmetric functional 0 ¤ ' 2 E  . Let L.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a nontrivial positive symmetric functional on E.M, /.

Section 4.9 Symmetric Functionals on Symmetric Operator Spaces

145

The case when M is an infinite atomless von Neumann algebra can be treated in a similar manner. The only difference is that the reference to Theorem 4.6.1 (c) has to be replaced with the reference to Theorem 4.6.1 (b) or Theorem 4.6.1 (a). Let E.M, / be a symmetric operator space on an atomic von Neumann algebra M and let E be the corresponding symmetric sequence space. It follows from the assumptions that E.M, / ¤ L1 .M, / or, equivalently, E ¤ l1 . By the assumptions, there exists an element x D .A/ 2 E such that m1m x 6! 0 in E. Let F be the symmetric function space constructed in Proposition 3.6.3. Since E ¤ l1 , it follows that F 6 L1 .0, 1/. Recall that the sequence space E is isometrically embedded into the function space F . We have x 2 F and m1 m x 6! 0 in F . By Theorem 4.6.1, there exists a positive symmetric functional 0 ' 2 F  . The restriction of the functional ' to E is a nontrivial positive symmetric functional on E. Let L.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a nontrivial positive symmetric functional on E.M, /. (d) ) (b) The proof is very similar to that of the implication (d) ) (a) and is, therefore, omitted. The only difference is that the references to Theorem 4.6.1 have to be replaced with references to Theorem 4.7.1. (d) ) (c) Firstly, we assume that the algebra M is -finite. Without loss of generality, .1/ D 1. Let E.M, / be a symmetric operator space and let E be the corresponding symmetric function space. By assumption, there exists an element x D .A/ 2 E such that m1m x 6! 0 in E. By Theorem 4.8.1, there exists a positive symmetric but not fully symmetric functional ' 2 E  . Let L.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a symmetric but not fully symmetric functional on E.M, /. The case when M is an infinite atomless von Neumann algebra can be treated in a similar manner. Let E.M, / be a symmetric operator space on an atomic von Neumann algebra M and let E be the corresponding symmetric sequence space. It follows from the assumptions that E.M, / ¤ L1.M, / or, equivalently, E ¤ l1 . By assumption, there exists an element x D .A/ 2 E such that m1 m x 6! 0 in E. Let F be the symmetric function space constructed in Proposition 3.6.3. Since E ¤ l1 , it follows that F 6 L1.0, 1/. Recall that the sequence space E is isometrically embedded into the function space F . We have x 2 F and m1m x 6! 0 in F . By Theorem 4.8.1, there exists a positive symmetric functional ' 2 F  and a function 0 y  x such that '.y/ > '.x/. Set z D E..y/jf.n  1, ngn2N /. Clearly, z 2 E and '.z/ D '.y/ > '.x/. Hence, the restriction of the functional ' to E is a positive symmetric functional on E that is not fully symmetric. Let L.'/ be the functional on E.M, / defined in Theorem 4.4.1. Clearly, L.'/ is a positive symmetric functional on E.M, / that is not fully symmetric. Corollary 4.9.3. Let E be a symmetric ideal of L.H /. The following conditions are equivalent.

146

Chapter 4 Symmetric Functionals

(a) There exist nontrivial continuous traces on E. (b) There exists an operator A 2 E such that lim

m!1

1 ˚m kA kE > 0. m

(4.23)

Proof. If E ¤ L1, then the assertion follows from Theorem 4.9.2 and Lemma 2.7.4. For E D L1 the assertion is obvious.

4.10 How Large is the Set of Symmetric Functionals? In the preceding section we fully characterized, in the atomless or atomic case, the symmetric operator spaces with nonempty sets of symmetric functionals. In this section, we examine the structure of the latter set when the symmetric operator space is equipped with a Fatou norm. Theorem 4.10.1. Let M be an atomless or atomic von Neumann algebra equipped with a faithful normal semifinite trace . Let E.M, / be a symmetric operator space equipped with a Fatou norm. One of the following mutually exclusive statements holds. (a) The space E.M, / does not admit a nontrivial symmetric functional. (b) The space E.M, / admits a (up to a constant) unique nontrivial symmetric functional and it is a faithful normal semifinite trace. (c) The set of Hermitian symmetric functionals on E.M, / is an infinite-dimensional Banach lattice. Proof. The set of Hermitian symmetric functionals on E.M, / is a Banach lattice (see Theorem 4.3.5). We need to show that its dimension is either 0 or 1 or 1. If M is atomless, then the assertion follows from Theorem 4.10.5 (see below) and Theorem 4.4.1. If M D l1 , then the assertion follows from Theorem 4.10.5 (see below) and Theorem 4.4.4. If M is atomic, then the assertion follows from Theorem 4.4.1. We now proceed with the proof of Theorem 4.10.5 which is the key (commutative) ingredient in the proof of Theorem 4.10.1. If E is a symmetric function space equipped with a Fatou norm then the following lemma proves that the mapping on E, x ! lim

m!1

is never additive.

1 k.m .x// .0,1/ kE m

147

Section 4.10 How Large is the Set of Symmetric Functionals?

Lemma 4.10.2. Let E D E.0, 1/ (or, E D E.0, 1/) be a symmetric function space equipped with a Fatou P norm and let 0 x 2 E. For every n 2 N, there exists a decomposition x D nkD1 xk , such that xk  0, 1 k n, and such that lim

m!1

1 1 k.m .x// .0,1/ kE D lim k.m .xk // .0,1/ kE , m!1 m m

1 k n.

Proof. Without loss of generality, x D .x/. Moreover, we can assume that supp.x/  .0, 1/. That is, x D .x/ .0,1/ . Fix m 2 N. If k ! 1, then we obtain m ..x/ . 1 , 1 / / " m ..x/ .0, 1 / / k m

m

almost everywhere. By the definition of a Fatou norm, it follows that km ..x/ . 1 , 1 / /kE ! km ..x/ .0, k m

1 m/

/kE .

For each m 2 N, select a number f .m/ > m, such that   1 km ..x/ .0, 1 / /kE . km ..x/ . 1 , 1 / /kE  1  f .m/ m m m

(4.24)

Set m0 D 1 and let mi :D f i .1/, i 2 N. Here, f i :D f ı    ı f (i times). It follows from the definition of mi C1 that   1 kmi ..x/ .0, 1 / /kE . (4.25) kmi ..x/ . 1 , 1 / /kE  1  miC1 mi mi mi Define the sets Ak , 1 k n, by setting  [  1 1 . , Ak :D mi C1 mi i Dkmod n

Set xk :D .x/ Ak , 1 k n. It is clear that [

Ak D .0, 1/

1kn

Pn

and, therefore kD1 xk D .x/ .0,1/ D x. If i D k mod n, then k.mi .xk // .0,1/ kE  kmi ..x/  It now follows from (4.25) that

1 1 miC1 , mi

 /kE .

  1 1 1 1 k.mi .x// .0,1/ kE . k.mi .xk // .0,1/ kE  mi mi mi

148

Chapter 4 Symmetric Functionals

Letting i D k mod n ! 1, we obtain mi ! 1 and, therefore, lim

m!1

1 1 k.m .xk // .0,1/ kE  lim k.m .x// .0,1/ kE . m!1 m m

The converse inequality is obvious. Lemma 4.10.3. Let E D E.0, 1/ 6 L1.0, 1/ be a symmetric function space equipped with a Fatou norm and P let 0 x 2 E be bounded. For every n 2 N, there exists a decomposition x D nkD1 xk such that xk  0, 1 k n, and such that 1 1 km .x/kE D lim km .xk /kE , 1 k n. lim m!1 m m!1 m Proof. Without loss of generality, x D .x/. Fix m > 0. If k ! 1, then we obtain that m ..x/ .0,k/ / " m .x/. By the definition of a Fatou norm, it follows that km ..x/ .0,k/ /kE ! km .x/kE . For each m 2 N, select a number f .m/ > m, such that   1 km .x/kE . km ..x/ .0,f .m// /kE  1  m

(4.26)

Set m0 D 0 and define the strictly increasing sequence mi , i 2 N, by setting mi :D .1/. Here, f i :D f ı    ı f (i times). We have f   1 kmi .x/kE , i  0. (4.27) kmi ..x/ .0,miC1 / /kE  1  mi i 1

Since E 6 L1.0, 1/, it follows that the fundamental function of E satisfies the condition .t / D o.t / as t ! 1. Passing to a subsequence (if needed), we preserve (4.27) and obtain that .mi mi C1/ 2i mi C1,

i  0.

(4.28)

For 1 k n, define the set Ak by setting [ .mi , mi C2 /. Ak :D i D2k mod 2n

Set xP k :D .x/ Ak , 1 k n. It is clear that [1kn Ak D .0, 1/ and, therefore, x D nkD1 xk .

149

Section 4.10 How Large is the Set of Symmetric Functionals?

For every 1 k n and every i D 2k mod n, we have kmiC1 .xk /kE  kmiC1 ..x/ .mi ,miC2 / /kE  kmiC1 ..x/ .0,miC2 / /kE  kmiC1 ..x/ .0,mi / /kE  kmiC1 ..x/ .0,miC2 / /kE  kxk1 

.mi mi C1/.

It follows from (4.27) and (4.28) that   1 kmiC1 .xk /kE  1  kmiC1 .x/kE  2i mi C1  kxk1. mi C1 Letting i ! 1, we obtain mi ! 1 and, therefore, lim

m!1

1 1 km .xk /kE  lim km .x/kE . m!1 m m

The converse inequality is obvious. Lemma 4.10.4. Let E D E.0, 1/ 6 L1.0, 1/ be a symmetric function space and let 0 x 2 E \ L1. We have lim

m!1

1 1 km .x/kE D lim k.m .x// .0,1/ kE . m!1 m m

Proof. Let be the (concave majorant of the) fundamental function of E. It follows from Theorem II.5.5 of [139] that kzkE kzkM  , It now follows that

8z 2 E.

 1 , x .0,1/ C .m .x// .1,1/ kE m   1 , x .0,1/ C .m .x// .1,1/ kM  k m   Z 1 1 0 ,x C m D .1/ .ms/.s, x/ds. m 1=m 

k.m .x// .1,1/ kE k

Therefore,

  Z 1 1 1 1 k.m .x// .1,1/ kE  ,x .1/ C m m m 1=m

We have 0

a.e.

.ms/.s, x/ .1=m,1/ .s/ ! 0

0

.ms/.s, x/ds.

150

Chapter 4 Symmetric Functionals

and 0

.ms/.s, x/ .1=m,1/ .s/ .s, x/.

It now follows from the Dominated Convergence Theorem that Z 1 0 .ms/.s, x/ds ! 0. 1=m

Since x 2 L1, it follows that .1=m, x/ D o.m1 / and, therefore, 1 k.m .x// .1,1/ kE ! 0. m The assertion now follows from the triangle inequality. Theorem 4.10.5. Let E be a symmetric function space equipped with a Fatou norm. One of the following mutually exclusive statements holds. (a) The space E does not admit a nontrivial symmetric functional. (b) The space E admits a (up to a constant) unique nontrivial symmetric functional, that is, an integral. (c) The space E admits an infinite number of linearly independent singular fully symmetric functionals. Proof. We consider only the case E D E.0, 1/ 6 L1 .0, 1/ (this automatically excludes (b)). All the remaining cases can be treated identically (using Lemma 4.10.2 instead of the combination of Lemma 4.10.2, Lemma 4.10.3 and Lemma 4.10.4). Suppose that there exists a symmetric functional on E. This excludes (a). Now, we show that (c) holds. By Theorem 4.2.2 there exists a positive symmetric functional on E. By Theorem 4.9.2 (or Theorem 4.6.1), there exists an element x D .x/ 2 E such that 1 lim km .x/kE > 0. m!1 m It follows that either lim

m!1

1 km ..x/ .0,1/ /kE > 0 m

or lim

m!1

1 km .minf.x/, .1/g/kE > 0. m

Without loss of generality we have either x D minf.x/, .1/g or x D .x/ .0,1/ .

Section 4.10 How Large is the Set of Symmetric Functionals?

151

Consider the Pnfirst case. By Lemma 4.10.3, there exist positive elements xk , 1 k n, such that kD1 xk D x and 1 1 km .xk /kE D lim km .x/kE > 0. m!1 m m!1 m lim

Consider the second case. By Lemma 4.10.2, there exist positive elements xk , 1 P k n, such that nkD1 xk D x and lim

m!1

1 1 k.m .xk // .0,1/ kE D lim k.m .x// .0,1/ kE > 0. m!1 m m

By Lemma 4.10.4, we have lim

m!1

1 1 km .xk /kE D lim km .x/kE > 0. m!1 m m

By Theorem 4.7.1, for every 1 k n, there exists a singular fully symmetric functional 'k such that k'k kE  1 and 1 1 km .xk /kE D lim km .x/kE > 0. m!1 m m!1 m

'k .xk / D lim

Since 'k is a positive functional, we have 'k .xk / 'k .x/ D

1 1 'k .m .x// km .x/kE . m m

Since m is arbitrarily large, we have 'k .x/ D 'k .xk / D lim

m!1

1 km .x/kE . m

Therefore, 'k .xl / D 0 for every l ¤ k. It clearly follows that functionals 'k , 1 k n, are linearly independent. Thus, for an arbitrarily large n  1, there exist n linearly independent fully symmetric functionals on E. Hence, the linear space of all fully symmetric functionals on E cannot be finite-dimensional. Therefore, it is infinite-dimensional. Since it is a closed subspace of a Banach space E  , then it is a Banach space itself and the assertion follows.

152

Chapter 4 Symmetric Functionals

4.11 Notes Symmetric Functionals The concept of a symmetric functional (as a positive continuous functional which takes the same value on those positive elements with the same distribution function) was introduced in 1998 in the paper [67]. This concept bears a strong resemblance to the Calkin correspondence between symmetrically normed ideals of compact operators and symmetric sequence spaces. Indeed, if we study whether symmetric (quasi-)norms can be lifted from commutative sequence and/or function spaces to their noncommutative counterparts, then it is natural also to ask whether linear symmetric functionals are also lifted. However, it should be noted that in the literature devoted to the study of singular functionals on Lorentz function spaces, as discussed in the end notes to Chapter 2, the concept of a symmetric functional was not introduced. Our first contact with such an object came from noncommutative geometry, from Alain Connes’ use of Dixmier traces and the subsequent exposure of Dixmier traces to a wide audience. The same term “symmetric functionals” was used by Figiel and Kalton [92] who studied a similar object (but without any reference to continuity or positivity). We refer to [92] for additional historical comments, and also for potential connections between symmetric functionals and analytic functions. Theorem 4.4.1 provides a natural bijection between the set of all symmetric functionals on a symmetric operator space E .M, / and the set of all symmetric functionals on the corresponding symmetric function space E, observed first for the case of the set of fully symmetric functionals in [67]. With regard to Theorem 4.6.2, which states that a symmetric functional decompose into normal and singular symmetric functionals, it was already established in [67, Proposition 2.2] that if E was a symmetric function space and 0 < ' 2 E  was a nontrivial R 1 normal and symmetric functional, then necessarily E  L1 .0, 1/ and '.f / D ˛ 0 f .s/ds for all f 2 E and some constant ˛ > 0. Question 4.1.2 was suggested in [65, 103, 104, 127]. Question 4.1.2 (c) was answered in the affirmative in [128] for the special case of a Lorentz ideal of compact operators. It should be pointed out that the method used in [128] cannot be extended to an arbitrary Lorentz ideal M .M, / and, furthermore, cannot be extended to a general symmetric operator space. The exposition in this chapter closely follows that of [239]. Our strategy in this chapter is based on the approach from the recent papers [240] and [130] (see also [20]), where condition (4.1) was connected to the geometry of E .M, /. The condition (4.1) is easy to verify in concrete situations. For example, the main result of [128] follows immediately from Theorem 4.1.3. Some examples of symmetric spaces, which are not Lorentz spaces, that admit symmetric traces were presented in [65]. These results are also an immediate corollary of Theorem 4.1.3. Theorem 4.10.1 is a response to questions of V. Kaftal and G. Weiss [119, 121] concerning the codimension of the commutator subspace (see the next chapter). They conjectured that it is always 0, or 1, or 1. The result in Theorem 4.10.1 confirms this conjecture for symmetric operator spaces, although in a rephrased form, since it confirms that the codimension of the closure of the commutator subspace, as studied in the next chapter, is 0, or 1, or 1.

Chapter 5

Commutator Subspace

5.1 Introduction In this chapter we look at the Lidskii formula for continuous traces on symmetrically normed ideals of compact operators. Let H be a separable Hilbert space. In this chapter we consider only the atomic factor M D L.H / of all bounded operators on H . Recall from Section 2.5 that symmetrically normed ideals are all the symmetric operator spaces for L.H /, and, from Section 2.7, that the set of continuous traces on a symmetrically normed ideal is the set of symmetric functionals on that ideal. According to Theorem 4.4.1, if E :D E.L.H /, Tr/ is a symmetrically normed ideal of compact operators and ' 2 E  is a continuous trace, then there exists a unique symmetric functional f 2 E  on the Calkin sequence space E such that '.A/ D f ı .A/,

0 A 2 E,

where .A/ is the sequence of singular values of the compact operator A. This identification can be extended to eigenvalues. Throughout this chapter .A/ denotes an eigenvalues sequence of a compact operator A (see Definition 1.1.10). Theorem 5.1.1 (Lidskii formula for continuous traces). If E is a symmetrically normed ideal of compact operators with Calkin sequence space E, and ' 2 E  is a continuous trace with corresponding symmetric functional f 2 E  , then any eigenvalue sequence .A/ 2 E for A 2 E and '.A/ D f ı .A/,

A 2 E.

This result is not trivial, as the technicalities about quasi-nilpotent operators in Section 5.5 demonstrate. Given the Lidskii formula, the bijective correspondence between continuous traces on a symmetrically normed ideal and symmetric functionals on the Calkin sequence space is unequivocal: ' D f ı  , f D ' ı diag, where diag is the diagonal operator for any orthonormal basis of H . Part III will examine the consequences of the Lidskii formula, and some variants of it, for Dixmier traces.

154

Chapter 5 Commutator Subspace

To obtain the Lidskii formula, we study the commutator subspace Com.E/ of a symmetrically normed ideal of compact operators E, and its closure Com.E/ in the norm topology of E. Definition 5.1.2. Let E be a symmetrically normed ideal of compact operators. The subspace Com.E/ :D Lin.ŒA, B : A 2 E, B 2 L.H // is called the commutator subspace of E. Its closure in the norm topology of E Com.E/ :D Lin.ŒA, B : A 2 E, B 2 L.H // is called the closed commutator subspace. The commutator subspace is the common kernel of all traces (meaning here all unitarily invariant linear functionals on E), and the closed commutator subspace is the common kernel of all continuous traces on E (this latter fact is shown in Section 5.3). That is, ' 2 E  is a continuous trace if and only if ' vanishes on Com.E/. Define the Cesàro operator C : l1 ! l1 by

 C.x/ :D

1 n 1 X xj , 1Cn nD0

x D fxj g1 j D0 2 l1 .

j D0

The commutator subspace and the invariance of the Calkin space to the Cesàro operator are related by the following noncommutative analog of the Figiel–Kalton theorem. It is the main result of this chapter. The Lidskii formula will follow from this result. Theorem 5.1.3. Let E be a symmetrically normed ideal of compact operators with Calkin sequence space E. If A 2 E, then .A/ 2 E and (a) A 2 Com.E/ if and only if C..A// 2 E. (b) A 2 Com.E/ if and only if   1 1 1  m C..A// ! 0 log.m/ m in the norm topology of E. Here, .A/ denotes an eigenvalue sequence of A. Part (a) is a known result from the combined work of Kalton and Dykema, Figiel, Weiss and Wodzicki. Part (b) is new. Section 5.6 discusses some corollaries of the

Section 5.2 Normal Operators in the Commutator Subspace

155

spectral identification in Theorem 5.1.3, including the proof of Theorem 5.1.1. Theorem 5.1.3 will also be pivotal to the applications in Part IV.

5.2 Normal Operators in the Commutator Subspace Let .E, k  kE / be a symmetrically normed ideal of compact operators with corresponding Calkin sequence space E. Theorem 5.1.3 is proved by two steps. The first step is lifting appropriate versions of the Figiel–Kalton theorem (one version for the commutator subspace and one version for the closed commutator subspace) from the sequence space E to the ideal of compact operators E. The lifting will provide the result of Theorem 5.1.3 for normal compact operators in E. The second step is to use Ringrose’s result (Theorem 1.1.22 in Chapter 1): if A is a compact operator then there exists a normal compact operator N and a compact quasi-nilpotent operator Q such that ADN CQ and A and N can have the same eigenvalue sequence, .A/ D .N /. We will show that if A belongs to a symmetrically normed ideal of compact operators E, then N (and hence Q) also belongs to E. Theorem 5.1.3 for an arbitrary A 2 E will follow from the first step when we show, in Section 5.5, that the quasi-nilpotent operator Q 2 Com.E/. The two steps above are the master plan for proving Theorem 5.1.3. In this section we show the statement of Theorem 5.1.3 (a) for normal operators. For the background of this result, which originated in [80], see the end notes of this chapter. Theorem 5.2.1. Let A 2 E be a normal operator and let .A/ 2 E be an eigenvalue sequence of A. We have A 2 Com.E/ if and only if C .A/ 2 E. We prove Theorem 5.2.1 using the following lemmas. Lemma 5.2.2. Let E be a symmetrically normed ideal of compact operators. For all positive operators A1 , A2 2 E such that .A1 / D .A2 /, we have A1 A2 2 Com.E/. In particular, 2A  A ˚ A 2 Com.E/. Proof. Let U be a partial isometry such that U  U D EA1 .0, 1/, U U  D EA2 .0, 1/ and UA1 U  D A2 . It follows that A1  A2 D A1  UA1 U  D U  UA1  UA1 U  D ŒU  , UA1  2 Com.E/. In particular, 2A  A ˚ A D .A  A ˚ 0/ C .A  0 ˚ A/ 2 Com.E/ C Com.E/ D Com.E/.

156

Chapter 5 Commutator Subspace

The next elementary lemma is stated (but not proved) in [178]. Lemma 5.2.3. Let  2 Rn be such that of the set f0, 1, : : : , n  1g such that sup 0kn1

Pn1

kD0 k

k ˇX ˇ ˇ ˇ .i /ˇ ˇ i D0

D 0. There exists a permutation 

sup 0kn1

jk j.

Proof. Without loss of generality, all the values k , 0 k n  1, are distinct. We may assume that .0, C/  .0,  /. There exists a unique number .0/ such that .0/ D .0, C /. Set n0 D 1. We use induction to define nk and .k/, 0 k P n  1. Suppose that ni and .i/, 0 i k, are already defined. If kiD1 .i / < 0, P then set .kC1/ :D .nk , C/ and nkC1 :D nk C 1. If kiD1 .i /  0, then set .kC1/ :D .k C 1  nk , C/ and nkC1 :D nk . Lemma 5.2.4. Let A 2 Mn .C/ be a self-adjoint operator such that Tr.A/ D 0. There exists an operator B 2 Mn .C/ and a partial isometry U 2 Mn .C/ such that A D ŒU , B and kBk1 kAk1 . Proof. Let k , 0 k n  1, be the eigenvalues of the operator A. Using the permutation constructed in Lemma 5.2.3, we can assume that k ˇ ˇX ˇ ˇ i ˇ kAk1 , ˇ

0 k n  1.

i D0

Select a basis in which A corresponds to the diagonal matrix Pn1diag.0, : : : , n1 /. Let Ei ,j , 0 i, j n  1, be matrix units. Set U :D i D1 Ei 1,i . We have A D ŒU , U  D with D D diag.0, 0 C 1 , : : : , 0 C    C n1 /. The claim follows by setting B :D U  D. Lemma 5.2.5. For every positive operator A 2 E and every positive operator B such that .B/ D E..A/jf0g [ fŒ2n , 2nC1/gn0 /, we have A  B 2 Com.E/. Proof. Let ek 2 H , k  0, be such that Aek D .k, A/ek . Without loss we may assume that Bek D .k, B/ek . Let pn be the projection on the linear span of ek , k 2 Œ2n , 2nC1 /. It is clear that Tr.pn .A  B/pn / D 0,

kpn .A  B/pn k1 .2n , A/.

Section 5.2 Normal Operators in the Commutator Subspace

157

By Lemma 5.2.4, there exist operators Cn D pn Cnpn and partial isometries Un D pn Un pn such that kCnk1 .2n , A/.

pn .A  B/pn D ŒUn , Cn , The series C D

1 X

Cn ,

U D

nD0

1 X

Un

nD0

converge strongly. Clearly, U is a partial isometry. We have   X M 1 1 n .Cn /  .2 , A/ Œ2n ,2nC1 / 2 .A/ .C / D  nD0

nD0

and, therefore, C 2 E. Hence, X X pn .A  B/pn D ŒUn , Cn  D ŒU , C  2 Com.E/. AB D n0

n0

Recall that for any two-sided ideal E, if A 2 E then A D .n,A/

Arguing similarly and using the normality of A, we have n ˇX ˇ .k, 0.

0

Set y D  0 2 L1 .0, 1/. Since y is constant on every interval .n, n C 1/, n  0, we obtain y 2 l1 . Observe that y  .A/ and y … l1 . By Theorem 7.5.2, y D fhAek , ek igk0 for some orthonormal basis fek gk0. Since y  .B/, and since ' is fully symmetric and '.B/ D 0, it follows from Theorem 4.4.1 that ' ı diag.y/ D 0. The first assertion is shown. Suppose A 2 M is positive such that ! .A/ D 0. By Lemma 7.5.1 and the fact that a Dixmier trace is a singular fully symmetric trace on M , we have that 0 ! ı diag.fhAek , ek igk0/ ! .A/ D 0. Formula (7.22) therefore holds. Conversely, if (7.22) holds for every orthonormal basis, then the first assertion proves that ! .A/ D 0. The second assertion is shown.

7.6 Notes Lidskii Formula The material in this chapter may be viewed as an attempt to extend the classical diagonal and Lidskii formulas to Dixmier traces. Detailed reference to Lidskii’s original 1959 result can be found in the notes to Chapter 1. The second author’s interest in this theme began with a meeting with Thierry Fack in Delft in 1998, at which, during informal discussions, Thierry stated the following important question: “is every Dixmier trace spectral?” The answer to this question was (independently and almost simultaneously) given in the papers [89] and [81, 82]. A proposition in [89] noted the formula in Theorem 7.3.1 for a classical pseudo-differential operator of order d on a closed d -dimensional Riemannian manifold, a topic we return to

243

Section 7.6 Notes

in Part IV. However, generally, the question how precisely to compute the value of a given Dixmier trace from the spectrum of a given operator remained open. The Lidskii formulas as shown in this chapter were noted in [6] under significant additional constraints on ! (for T 2 L1,1 ). Theorem (from [6]). Let ! be M -invariant and let T 2 L1,1 . We have   X 1  , ! .T / D ! log.n/ jj>1=n,2 .T /

where .T / is the spectrum of T . In the case when T is a positive arbitrary element from M1,1 and ! is taken from a rather special subset of all M -invariant extended limits (termed in [29] “maximally invariant Dixmier functionals”), this result can already be found in [32, Proposition 2.4]. In [9, Theorem 1], the assertion from [32, Proposition 2.4] was extended to an arbitrary M -invariant !. Another modification of the class of extended limits for which the results [32, Proposition 2.4] and [9, Theorem 1] hold is given in [33, Proposition 4.3]. The spectral counting formula in the form of the radius .n/=n was achieved in [215]. The approach in [215] forms the basis of this chapter.

Chapter 8

Heat Kernel Formulas and -function Residues

8.1 Introduction The interplay between Dixmier traces, -function residues and heat kernel formulas is a cornerstone of noncommutative geometry [48]. These formulas are widely used in physical applications. To define these objects, fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace . Let A and B be positive operators from M. Consider the following Œ0, 1-valued

-functions s ! .A1Cs /,

s ! .A1Cs B/.

(8.1)

Clearly, these mappings are defined for s D 0 if and only if A 2 L1 .M, /. This chapter shows (see Lemma 8.6.2) that .A1Cs / D O.s 1 / if A 2 M1,1 .M, /, where M1,1 .M, / is the Dixmier–Macaev operator ideal associated to the pair .M, /, see Example 2.6.10. It therefore makes sense to ask about a residue of the so defined function at s D 0. The latter can be defined using extended limits by setting 

 1 1C1=t

 .A/ :D .A / , t



 1 1C1=t

,B .A/ :D .A B/ , t

(8.2)

where is an extended limit on L1 .0, 1/. Consider the following Œ0, 1-valued heat kernel functions. Set s ! .exp.sA1 //,

s ! .exp.sA1 /B/.

(8.3)

This chapter shows that .exp.sA1 // D O.s 1 / if A 2 L1,1 .M, / (here we recall that L1,1 .M, / is the smaller operator ideal inside the Dixmier–Macaev operator ideal, consisting of those operators A 2 M such that .t , A/ D O.t 1/). It is also shown that the functions     1 1 1 1 (8.4) M t ! .exp..tA/ // , M t ! .exp..tA/ //B t t belong to L1 .0, 1/ if A 2 M1,1 .M, /. Here M is the logarithmic mean. It therefore makes sense to ask about a residue of the so defined heat kernel function at s D 0.

Section 8.1 Introduction

The latter can be defined using extended limits by setting   1 1  .A/ :D . ı M / .exp..tA/ // , t   1 1 ,B .A/ :D . ı M / .exp..tA/ //B , t

245

(8.5) (8.6)

where is an extended limit on L1 .0, 1/. We prove in Section 8.2 that if ! is a dilation invariant extended limit, then the heat kernel functional ! is a linear functional on M1,1 .M, /. In fact, we show in Proposition 8.2.5 that if ! is any extended limit such that ! is linear on M1,1 .M, /, then necessarily there exists a dilation invariant extended limit !0 such that ! D !0 . Moulay-Tahar Benameur and Thierry Fack, in [9], suggested a more general approach to heat kernel formulas. It consisted of replacing the function t ! exp.t 1 / with an arbitrary Schwartz function f . The following equality was proved in [9],  Z 1    1 1 ds  ! .AB/ f (8.7) ! .f .tA/B/ D t s 0 for A 2 L1,1 .M, / under the assumption that ! is M -invariant (that is, ! D ! ıM ). In Theorem 8.5.1 below, we show that (8.7) holds for any dilation invariant extended limit !, for any twice-differential bounded function f on the semi-axis such that f .0/ D f 0 .0/ D 0, and for any A 2 M1,1 .M, /. The main result of this chapter is the connection of heat kernel formulas and function residues to Dixmier traces. We recall, from the methods of Chapter 6, that a Dixmier trace ! can be defined on the Dixmier–Macaev operator ideal M1,1 .M, / by lifting, as in Section 4.4, a Dixmier trace from the Lorentz function space M1,1 , and that the set of Dixmier traces coincides with the set of all normalized fully symmetric functionals on M1,1 .M, /. Here ! is any dilation invariant extended limit on L1 .0, 1/ (see Theorem 6.3.6 and the notes to Chapter 6) and we recall that a fully symmetric functional ' 2 M1,1 .M, / is normalized if '.A/ D 1 whenever .t , A/ D .1 C t /1 , t > 0 (atomless case), or .n, A/ D .1 C n/1 , n  0 (atomic case). It is proved in Theorem 8.2.4 and Theorem 8.6.4 that ! and  , when linear, are normalized fully symmetric functionals on M1,1 .M, /. It is therefore quite natural to ask whether every normalized fully symmetric functional on M1,1 .M, / is of the form ! (or  , respectively). Firstly, in Theorem 8.2.4 we prove that if ! is dilation invariant, then the functional ! extends to a normalized fully symmetric functional on M1,1 .M, /. Secondly, in Theorem 8.3.6 we show that in fact every normalized fully symmetric functional on M1,1 .M, / coincides with some ! , where ! is dilation invariant. Thus, we can conclude that the set of functionals ! where ! is dilation invariant exactly coincides with the set of all Dixmier traces. Theorem 8.2.9 shows that ! D !

246

Chapter 8 Heat Kernel Formulas and -function Residues

if ! is an M -invariant extended limit on L1 .0, 1/. The theorem also shows that the question as to whether the equality ! D ! holds for every dilation invariant extended limit ! is answered in the negative. Surprisingly, the analogous results fail for -functions. Even though Theorem 8.6.8 confirms that ! D !ılog , for those dilation invariant extended limits ! such that ! ı log is also dilation invariant, Section 8.7 shows that the set of fully symmetric functionals of the form  , is an extended limit on L1 .0, 1/, is strictly smaller than the set of all normalized fully symmetric functionals. Hence, the -function residues provide a smaller set of traces on M1,1 .M, / than either the heat kernel formulas or the Dixmier traces.

8.2 Heat Kernel Functionals Fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace . Let M1,1 .M, / be the Dixmier–Macaev operator ideal associated to the pair .M, /, that is M1,1 .M, / :D fA 2 M : .A/ 2 M1,1 g , kAkM1,1 :D k.A/kM1,1 , where M1,1 is the Lorentz function space  M1,1 :D x 2 L1 .0, 1/ : kxkM1,1 :D sup t >0

1 log.1 C t /

Z

t

 .s, x/ds < 1 ,

0

see also Example 2.6.10. Throughout this chapter the symbol M : L1 .0, 1/ ! L1 .0, 1/ denotes the logarithmic mean Z t 1 ds .M x/.t / :D x.s/ , x 2 L1 .0, 1/, t > 0. log.t / 1 s Definition 8.2.1. For every extended limit ! on L1 .0, 1/, the functional ! : M1,1 .M, /C ! R defined by setting   1 .tA/1 / , 0 A 2 M1,1 .M, /, ! .A/ :D .! ı M / .e t is called a heat kernel functional. The following lemma is a variant of the Jensen inequality. We refer the reader to [98] for its proof (see Lemma II.3.4 there). Lemma 8.2.2. Let F : RC ! RC be a convex function such that F .0/ D 0. If A, B 2 M are positive operators such that B  A, then .F .B// .F .A//. In particular, F .A/ 2 L1 .M, / implies that F .B/ 2 L1 .M, /.

247

Section 8.2 Heat Kernel Functionals

The following lemma shows that a heat kernel functional is well defined. Lemma 8.2.3. If 0 A 2 M1,1 .M, /, then   1 .tA/1 .e M / 2 L1 .0, 1/. t For every extended limit ! on L1 .0, 1/, we have   1 .tA/1 .Ae / . ! .A/ D ! log.1 C t / Proof. We break the proof into a few steps. (a) Let 0 A, C 2 M1,1 .M, / be such that .s, C / D kAkM1,1 =.1 C s/, s > 0. 1 The mapping z ! ze z , z > 0, is convex. Hence, by Lemma 8.2.2, Z 1 1 1 1 .Ae .tA/ / .C e .t C / / D .s, C e .t C / /ds. 0

It now follows from Corollary 2.3.17 (d) that Z 1 Z .tA/1 .t.s,C //1 / .s, C /e ds D kAkM1,1 .Ae 0

1

Therefore, Ae .tA/

e z dz . z .t kAkM1,1 /1

1

2 L1 and .Ae .tA/ / D O.log.1 C t //, t > 0.

(b) By definition of M , we have   Z t 1 1 1 ds .tA/1 .e / D .e .sA/ / 2 . M t log.t / 1 s Note that Z

t 0

1 ds e .sA/ 2 s

Z

1

D

1

1

e uA du D Ae .tA/ ,

t > 0.

1=t

Hence,    1 1 .tA/1 .tA/1 A1 .e .Ae / D /  .Ae / . M t log.t / 

(c) The right-hand side of the last equality in (b) is a continuous (even differentiable) function of t > 0. The continuity at t D 0 is obvious. Thus, it is locally bounded. The first assertion of the lemma now follows from (a). The second assertion follows from the final display above.

248

Chapter 8 Heat Kernel Formulas and -function Residues

Theorem 8.2.4. For every dilation invariant extended limit ! on L1 .0, 1/, ! extends to a fully symmetric linear functional on M1,1 .M, /. Proof. By Lemma 8.2.3, the functional ! is well defined on M1,1 .M, /C . We break the proof into steps. (a) Let 0 A, B 2 M1,1 .M, / be such that .B/ D 21=2.A/. We have 1

1

.Be .tB/ / D .Ae .2tA/ /,

t > 0.

It now follows from Lemma 8.2.3 that     1 1 1 1 ! .B/ D ! .Be .tB/ / D ! .Ae .2tA/ / . log.1 C t / log.1 C 2t / Since ! is dilation invariant, it follows that   1 .tA/1 / D ! .A/. .Ae ! .B/ D .! ı 1=2 / log.1 C t / 1

(b) Let 0 A, B 2 M1,1 .M, /, be such that B  A. The mapping z ! ze z , z > 0, is convex. Hence, by Lemma 8.2.2, 1

1

.Be .tB/ / .Ae .tA/ /. By Lemma 8.2.3, ! .B/ ! .A/ for every extended limit !. (c) Fix positive operators A, B 2 M1,1 .M, /. By Theorem 3.3.3 and Theorem 3.3.4, we have A ˚ B  A C B  21=2 .A ˚ B/. It now follows from (a) and (b) that ! .A C B/ D ! .A ˚ B/ D ! .A/ C ! .B/. (d) By (c), ! is an additive functional on M1,1 .M, /C . Since ! is also monotone, it follows that it extends to a linear functional on M1,1 .M, /. The latter is fully symmetric by (b). It appears to be a difficult task to describe the set of all extended limits ! for which Definition 8.2.1 defines a linear functional ! . The following theorem explains why we can restrict consideration to the dilation invariant extended limits.

249

Section 8.2 Heat Kernel Functionals

Theorem 8.2.5. If an extended limit ! on L1 .0, 1/ is such that ! is a linear functional on M1,1 .M, /, then there exists a dilation invariant extended limit !0 such that ! D !0 . Proof. Define the mapping S : M1,1 .M, /C ! L1 .0, 1/ by setting .SA/.t / :D

1 1 .Ae .tA/ /, log.1 C t /

0 A 2 M1,1 .M, /C .

Let Y be the linear span of the elements SA, where 0 A 2 M1,1 .M, /, and L01 .0, 1/. We have s 1 .SA/ 2 s 1 S.sA/ C L01 .0, 1/,

0 A 2 M1,1 .M, /.

Therefore, Y is a dilation invariant linear subspace of L1 .0, 1/. Suppose ! generates a linear functional on M1,1 .M, /. In particular, ! .sA/ D s! .A/ for every s > 0. It follows from Lemma 8.2.3 that, for 0 A 2 M1,1 .M, /, !.SA/ D ! .A/ D s 1 ! .sA/ D s 1 !.S.sA// D .! ı s 1 /.SA/. Thus, !jY is a dilation invariant linear functional. By the definition of an extended limit, !jY lim sup . Thus, the subspace Y , the dilation semigroup s , s > 0, the linear functional !jY and the convex functional lim sup on L1 .0, 1/ satisfy the conditions of Theorem 6.2.5. Thus, by Theorem 6.2.5, !jY extends to a dilation invariant extended limit !0 on L1 .0, 1/. In particular, ! D !0 . The next lemma is elementary. We include a detailed computation for completeness. Lemma 8.2.6. For every positive operator A 2 M and t > 0, we have     1 1 1 1 1 A C 3Ae .tA=2/ 4Ae .tA/ A  C 3Ae .2tA/ . t C t C Proof. We claim that .1  s/C C 3e 2s 4e s .1  s/C C 3e s=2 ,

s  0.

Indeed, if s  1, then the inequality is obvious. Hence, it is sufficient to verify that 3e 2s  4e s  s 1 3e s=2  4e s  s,

s 2 Œ0, 1.

Since the left-hand (respectively, right-hand) side of this inequality is convex (respectively, concave) on the interval Œ0, 1 and since the inequality holds for s D 0 and for s D 1, it follows that the inequality remains valid for s 2 Œ0, 1. This proves the claim.

250

Chapter 8 Heat Kernel Formulas and -function Residues

Substituting s D .t /1 , we obtain     1 1 1 .t =2/1 .t /1 C 3e 4e  C 3e .2t / .  t C t C The assertion follows from the Spectral Theorem (see Theorem 2.1.5). The following lemma delivers a simpler expression for ! . It is the crucial component of the proofs in the following sections. Lemma 8.2.7. For every 0 A 2 M1,1 .M, / and for every dilation invariant extended limit ! on L1 .0, 1/, we have     1 1 . (8.8)  A ! .A/ D ! log.1 C t / t C Proof. Denote, for brevity, the right-hand side of (8.8) by ! .A/. It follows from Lemma 8.2.6 and Lemma 8.2.3 that   1 3 ! .A/ C 6! A 4! .A/ ! .A/ C ! .2A/. 2 2 Since the extended limit ! is dilation invariant, then it follows from Theorem 8.2.4 that ! .sA/ D s! .A/ for every s > 0 and, therefore, ! .A/ D ! .A/. The next lemma concerns the behavior of the distribution function nA of an operator A 2 M1,1 .M, /. Lemma 8.2.8. For every 0 A 2 M1,1 .M, / and every dilation invariant extended limit ! on L1 .0, 1/, we have    1 1 nA D 0. ! t log.1 C t / t Proof. It follows from the dilation invariance of ! that         1 1 1 1 D! .  A  A ! log.1 C t / t C log.1 C 2t / 2t C On the other hand, we have log.1 C 2t / D .1 C o.1//  log.1 C t / and, therefore,         1 1 1 1 D! .  A  A ! log.1 C 2t / 2t C log.1 C t / 2t C Therefore,

 !

1  log.1 C t /

     1 1 D 0.  A A 2t C t C

Section 8.2 Heat Kernel Functionals

251

Since            

 1 1 1 1 1 1 1 1 1 D nA  A C A EA ,  nA ,  A 2t C t C 2t t t 2t t 2t t the assertion follows immediately. We are ready to present the last main result of this section. For any dilation invariant extended limit ! on L1 .0, 1/, define a Dixmier trace ! on M1,1 .M, / by linear extension of the formula   Z t 1 .s, A/ds , 0 A 2 M1,1 .M, /. ! .A/ D ! log.1 C t / 0 By the results of Chapter 6, and comments in the end notes of that chapter, ! is a continuous singular trace on M1,1 .M, / and, if M is atomless, or atomic, then the set of Dixmier traces ! , ! is a dilation invariant extended limit, coincides with the set of all normalized fully symmetric functionals on M1,1 .M, /. The next theorem links Dixmier traces on the Dixmier–Macaev operator ideal to heat kernel formulas, and also shows the limitations to their direct connection. This result will be an important part of the proof of Corollary 8.5.2. Theorem 8.2.9. If ! is a dilation invariant extended limit on L1 .0, 1/ such that ! D ! ı M (or ! D ! ı Ps , s > 0), then ! D ! . There exists a dilation invariant extended limit ! on L1 .0, 1/ such that ! ¤ ! . Proof. Let A 2 M1,1 .M, / be an arbitrary positive operator. By the spectral theorem, we have      Z 1 1 1 1 1 1 D  A nA . ud nA .u/ C log.1 C t / t C log.1 C t / 1=t t log.1 C t / t It now follows from Lemma 8.2.7 and Lemma 8.2.8 that   Z 1 1 ! .A/ D ! ud nA .u/ . log.1 C t / 1=t The first assertion now follows from Theorem 7.4.3. The second assertion follows from Theorem 7.4.7.

252

Chapter 8 Heat Kernel Formulas and -function Residues

8.3 Fully Symmetric Functionals are Heat Kernel Functionals It follows from Theorem 8.2.4 that the heat kernel functional ! defined in Definition 8.2.1 is a normalized fully symmetric functional on M1,1 .M, / whenever ! is a dilation invariant extended limit on L1 .0, 1/. In this section, we show the converse. The statement of Lemma 8.2.7 suggests to define a (nonlinear) operator T : M1,1 .M, /C ! L1 .0, 1/ by the formula .TA/.t / :D

1  log.1 C t /

 A

1 t

  ,

t > 0.

(8.9)

C

We need some properties of the operator T . Firstly, we show that it is additive on certain pairs of operators 0 A, B 2 M1,1 .M, /. Lemma 8.3.1. For every pair of positive operators A, B 2 M1,1 .M, /, we have T .A ˚ B/ D TA C TB. Proof. Immediately from the assumption we have for t > 0 that       1 1 1 A˚B  D A C B . t C t C t C The result follows due to the linearity of . Corollary 8.3.2. The set Y D T M1,1 .M, /C  T M1,1 .M, /C is a dilation invariant (not necessarily closed) linear subspace of L1 .0, 1/. Proof. If z1 , z2 2 Y , then there exist positive operators A, B, C , D 2 M1,1 .M, / such that z1 D TA  TB and z2 D T C  TD. By Lemma 8.3.1, we have z1 C z2 D TA C T C  TB  TD D T .A ˚ C /  T .B ˚ D/ 2 Y . For every  > 0, we have that  .A/,  .B/ are positive elements from M1,1 and therefore z1 D TA  TB D T . .A//  T . .B// 2 Y . Similarly, for every  < 0, we have z1 D jjTB  jjTA D T .jj .B//  T .jj .A// 2 Y . Thus, Y is a linear subspace of L1 .0, 1/.

Section 8.3 Fully Symmetric Functionals are Heat Kernel Functionals

253

It follows from the definition (8.9) that for every positive A 2 M1,1 .M, / and for every t , s > 0, we have         t 1 s s 1 1 .TA/ D  A  s A D . s log.1 C t =s/ t C log.1 C t =s/ t C Therefore, s .TA/ 2 sT .s 1 A/ C L01 .0, 1/. Hence, the subspace Y is dilation invariant. Next, we explain the connection of the operator T with fully symmetric functionals on M1,1 .M, /. We continue to use the notation introduced in Corollary 8.3.2. Lemma 8.3.3. For every symmetric functional ' on M1,1 .M, /, there exists a unique linear functional : Y ! R such that ' D ı T on M1,1 .M, /C . Proof. If z 2 Y is such that z D TA  TB for 0 A, B 2 M1,1 .M, /, then set .z/ :D '.A/  '.B/. First, we prove that is well defined. Let z 2 Y be such that z D TA  TB and z D T C  TD for 0 A, B, C , D 2 M1,1 .M, /. It follows from the definition of T and Lemma 8.3.1 that T .A ˚ D/ D TA C TD D TB C T C D T .B ˚ C /. Thus, A ˚ D is equimeasurable with B ˚ C . It follows that '.A/ C '.D/ D '.A ˚ D/ D '.B ˚ C / D '.B/ C '.C /. Hence, '.A/  '.B/ D '.C /  '.D/. This proves that is a well-defined functional. Next, we prove the linearity of . Let z1 D TA  TB and let z2 D T C  TD. It follows from the definition of T and Lemma 8.3.1 that z1 C z2 D TA C T C  TB  TD D T .A ˚ C /  T .B ˚ D/. Therefore, .z1Cz2 / D '.A˚C /'.B˚D/ D .'.A/'.B//C.'.C /'.D// D .z1/C .z2 /.

254

Chapter 8 Heat Kernel Formulas and -function Residues

If   0, then .z1/ D .T ..A//  T . .B/// D '. .A//  '. .B// D .'.A/  '.B// D  .z1/. Similarly, if  0, then .z1/ D .T .jj.B//  T .jj .A/// D '.jj .B//  '.jj .A// D jj.'.B/  '.A// D  .z1/. Hence, the functional : Y ! R is linear. Lemma 8.3.4. Let 0 A, B 2 M1,1 .M, / be such that TB TA. For every fully symmetric functional ' on M1,1 .M, /, we have '.B/ '.A/. Proof. It follows immediately from the definition (8.9) that the assumption TB TA is equivalent to       1 1  B  A , 8t > 0. t C t C Applying Theorem 3.3.6 we obtain B  A and, therefore, '.B/ '.A/. Lemma 8.3.5. Let 0 A, B 2 M1,1 .M, /. For every normalized fully symmetric functional ' on M1,1 .M, /, we have '.B/  '.A/ lim sup.TB  TA/.t /. t !1

(8.10)

Proof. Let 0 C 2 M1,1 .M, / be an operator such that .t , C / D .1 C t /1 , t > 0. Let c be an arbitrary real number greater than the right-hand side of (8.10). It follows from the definition of T that .T .jcjC //.t / 2 jcj C L01 .0, 1/. We split the proof into the cases c  0 and c < 0. If c  0, then it follows from Lemma 8.3.1 that lim sup.TB  T .A ˚ cC //.t / D c C lim sup.TB  TA/.t / < 0. t !1

t !1

Hence, there exists N > 0 such that we have .TB/.t / .T .A ˚ cC //.t /,

8t > N .

255

Section 8.3 Fully Symmetric Functionals are Heat Kernel Functionals

It now follows from the definition of T : M1,1 .M, /C ! L1 .0, 1/ that    1 T .A ˚ cC /. T min B, N Since ' is singular and fully symmetric, it follows from Lemma 8.3.4 that    1 '.B/ D ' min B, '.A ˚ cC / D c C '.A/. N Since c is an arbitrary number exceeding the right-hand side, the inequality (8.10) follows. If c < 0, then it follows from Lemma 8.3.1 that lim sup.T .B ˚ cC /  TA/.t / D c C lim sup.TB  TA/.t / < 0. t !1

t !1

Hence, there exists N > 0 such that we have .T .B ˚ cC //.t / .TA/.t /,

8t > N .

It now follows from the definition of T : M1,1 .M, /C ! L1 .0, 1/ that    1 T .A/. T min B ˚ cC , N By Lemma 8.3.4, we have 



1 '.B/ D c C '.B ˚ cC / D c C ' min B ˚ cC , N

 c C '.A/.

Since c is an arbitrary number exceeding the right-hand side, the inequality (8.10) follows. The following theorem is the main result of this section. It shows that an arbitrary fully symmetric functional ' on M1,1 .M, / can be represented (up to a scalar) as a heat kernel formula. Theorem 8.3.6. Let ' be a normalized fully symmetric functional on M1,1 .M, /. There exists a dilation invariant extended limit ! on L1 .0, 1/ such that ' D ! . Proof. Let Y be the linear subspace of L1 .0, 1/ defined in Corollary 8.3.2 and let : Y ! R be the linear functional constructed in Lemma 8.3.3. By Lemma 8.3.5, we have .z/ lim sup.z/ for every z 2 Y . By the Hahn–Banach theorem, is an extended

256

Chapter 8 Heat Kernel Formulas and -function Residues

limit on L1 .0, 1/ (restricted to the subspace Y ). Arguing as in Corollary 8.3.2, for s > 0 we obtain that s .TA/ 2 sT .s 1 A/ C L01 .0, 1/. Since the extended limit vanishes on L01 .0, 1/, it follows that .s .TA// D s .T .s 1 A// D s'.s 1 A/ where the last equality uses Lemma 8.3.3. Since ' is a fully symmetric functional then s'.s 1 A/ D '.A/ D .TA/ where the last equality uses Lemma 8.3.3 again. Thus, .s z/ D .z/, for all z 2 Y . By Theorem 6.2.5, there exists a dilation invariant extended limit ! on the space L1 .0, 1/ such that ! D on Y . In particular, !.TA/ D .TA/ D '.A/, for all 0 A 2 M1,1 .M, /. By the definition of T , we have     1 1  A , 80 A 2 M1,1 .M, /. '.A/ D ! log.1 C t / t C The assertion now follows from Lemma 8.2.7.

8.4 Generalized Heat Kernel Functionals Let ! be an extended limit on L1 .0, 1/ and let B 2 M. Following [9], we consider the functionals on M1,1 .M, /C defined by the formula   1 (8.11) !,B,f .A/ :D .! ı M / t ! .f .tA/B/ , A  0 t where f is a function on the semi-axis. This section shows the validity of the definition (8.11) for a twice-differentiable bounded function f such that f .0/ D f 0 .0/ D 0. Theorem 8.4.1. Let f 2 C 2 Œ0, 1/ be a bounded function such that f .0/ D f 0 .0/ D 0. Let 0 A 2 M1,1 .M, / and let B 2 M. We have   1 M t ! .f .tA/B/ 2 L1 .0, 1/. t To prove the theorem we require the following lemmas.

257

Section 8.4 Generalized Heat Kernel Functionals

Lemma 8.4.2. If 0 A 2 M1,1 .M, /, then we have    (a)  A  1t C D O.log.t // as t ! 1. (b) 



˚

2    min A, 1t D O 1t log.t / as t ! 1.

Proof. Let c :D kAkM1,1 and set z.s/ D 1=.1 C s/, s > 0. We have A  cz. (a) It follows from Theorem 3.3.6 that       1 1  cz  D O.log.t //  A t C t C as t ! 1. (b) For fixed t > 0, we have  



1 min A, t

2 

  X  

 1 1 1 1 2  A EA nC1 , n D 2 nA C t t 2 t 2 t n0   X  

 1 1 1 1 1  EA nC1 , n C 2 nA t t 22n t 2 2 t 2 t n0    X  1 1 1 1 C nA nC1 . 2 nA t t 22n t 2 2 t n0

We have .s, A/ c log.1 C s/=s for every s > 0. Therefore,     1 1 c log.1 C s/  nA m s: 2cu log.u/ u s u for all sufficiently large u. It follows that  

     X 2c log.t / 1 2 2n nC1 2 .t log.t / C 2 log.2 t // D O . min A, t t t n0

Lemma 8.4.3. If f0 .t / :D minf1, t 2g, t > 0, then   1 .f0 .tA// 2 L1 .0, 1/, 0 A 2 M1,1 .M, /. t !M t Proof. For a fixed t > 0, we have    Z t Z t 1 1 1 2 2 2 2 M .f0 .tA// D .minfA , s g/ds D minfA , s gds .  t log.t / 1 log.t / 1

258

Chapter 8 Heat Kernel Formulas and -function Residues

If  > t 1 , then Z

t

minf2, s 2 gds D

Z

0

1

Z 2ds C

0

If  t 1 , then

Z

t

minf2, s 2 gds D

0

Z

t

t 1

ds 1 D 2  . 2 s t

2ds D 2 t .

0

In either case, we have Z

t 0

     1 1 2 minf2, s 2 gds D 2   C t min , . t C t

It follows from the Spectral Theorem that Z

t

2

minfA , s 0

2

     1 1 2 gds D 2 A  C t min A, . t C t

In particular,            1 1 2 t 1 2 1  A  min A, . C CO M .f0 .tA// D t log.t / t C log.t / t log.t / The assertion follows from Lemma 8.4.2. We can now prove Theorem 8.4.1. Proof of Theorem 8.4.1. Set f0.t / :D minf1, t 2g, t > 0. Observe that the assumptions on f guarantee that there exists a constant c > 0 such that jf .t /j cf0 .t /. It is now immediate that j.f .tA/B/j kBk1 .jf .tA/j/ ckBk1 .f0 .tA//. The assertion follows from Lemma 8.4.3.

8.5 Reduction of Generalized Heat Kernel Functionals The last section defined the generalized heat kernel functional for a dilation invariant extended limit ! on L1 .0, 1/, an arbitrary operator B 2 M, and a bounded function f 2 C 2Œ0, 1/ such that f .0/ D f 0 .0/ D 0,   1 !,B,f .A/ :D .! ı M / t ! .f .tA/B/ , 0 A 2 M1,1 .M, /. t

259

Section 8.5 Reduction of Generalized Heat Kernel Functionals

This section shows that this general heat kernel functional can be reduced to a scalar multiple of the weighted heat kernel functional ! .B/ from Section 8.2. Theorem 8.5.1. Let f 2 C 2 Œ0, 1/ be a bounded function such that f .0/ D f 0 .0/ D 0. Let 0 A 2 M1,1 .M, / and let B 2 M. For every dilation invariant extended limit ! on L1 .0, 1/ we have  Z 1 ds f .s/ 2 ! .AB/. (8.12) !,B,f .A/ D s 0 The proof of the theorem is given below. The following corollary treats the case of classical heat kernel formulas and their connection to Dixmier traces. Corollary 8.5.2. Let 0 A 2 M1,1 .M, / and let B invariant extended limit ! on L1 .0, 1/ we have    1 q .! ı M / .exp..tA/ /B/ D  1 C t

2 M. For every dilation  1 ! .AB/. q

If, in addition, ! D ! ı M , then     1 1 ! .AB/. .! ı M / .exp..tA/q /B/ D  1 C t q Here  is the Gamma function and q > 0. Proof. Use f : s ! exp.s q /, s > 0, in Theorem 8.5.1 and observe that   Z 1 ds 1 f .s/ 2 D  1 C . s q 0 The first assertion follows from Theorem 8.5.1. Combining this with Theorem 8.2.9, we obtain the second assertion. The following lemmas are used to prove Theorem 8.5.1. Lemma 8.5.3. If 0 A 2 M1,1 .M, / and B 2 M, then     1 1 B D ! .AB/ !  A log.1 C t / t C for every extended limit !.

(8.13)

260

Chapter 8 Heat Kernel Formulas and -function Residues

Proof. Recall that the function u ! .u  1=t /C is convex. It follows from Theorem A.2.7 in the appendix that       (a)  A  1t C B   B 1=2 AB 1=2  1t C if 0 B 1.       (b)  A  1t C B  B 1=2 AB 1=2  1t C if B  1. Hence, we have         1 1 1 1 !  A  B 1=2 AB 1=2  B ! log.1 C t / t C log.1 C t / t C (8.14) for 0 B 1 and         1 1 1 1 1=2 1=2  A  B AB  ! B ! log.1 C t / t C log.1 C t / t C (8.15) for B  1. Since both sides are homogeneous, the inequality (8.14) is valid if B  0, while the inequality (8.15) is valid if B is bounded from below by a strictly positive constant. Thus, we have the equality (8.13) valid for every B bounded from below by a strictly positive constant. Set Bn :D maxfB, 1=ng, n  1. It follows that equality (8.13) holds with B replaced with Bn throughout. Clearly, ˇ        ˇ ˇ ˇ 1 1 1 1 1 ˇ! Bn  ! B ˇˇ ! .A/.  A ˇ log.1 C t /  A  t log.1 C t / t C n C Therefore,         1 1 1 1  A  A B D lim ! Bn ! n!1 log.1 C t / t C log.1 C t / t C D lim ! .ABn / D ! .AB/. n!1

Lemma 8.5.4. If 0 A 2 M1,1 .M, / and B 2 M, then      s 1 .! ı M /  EA , 1 B D s 1 ! .AB/ t t for every dilation invariant extended limit ! and for every s > 0. Proof. We have           Z t 1 1 du s s M  EA , 1 B D ,1 2 B . EA t t log.t / 1 u u

Section 8.5 Reduction of Generalized Heat Kernel Functionals

However,

Z



t

EA 0

261

   du s 1 1 ,1 2 D s A  . u u t C

Therefore,          1 1 s 1 D!  s 1 A  .! ı M /  EA , 1 B B . t t log.1 C t / t C Since ! is dilation invariant, the assertion follows from Lemma 8.5.3. Lemma 8.5.5. Let f : RC ! R be supported and monotone on .a, b. Let 0 A 2 M1,1 .M, / and let B 2 M. For every dilation invariant extended limit ! on L1 .0, 1/ we have Z !,B,f .A/ D

b

f .s/ a

 ds ! .AB/. s2

Proof. Without loss of generality, we may assume that f is increasing on .a, b and that B  0. Let a D a0 a1 a2    an D b. Since f is increasing on .a, b, we have for every t > 0 n1 X

 f .ak /EA

kD0



 n1 X ak akC1 ak akC1 , f .tA/ , . f .akC1 /EA t t t t kD0

Therefore, !,B,f .A/

n1 X kD0

    ak akC1 1 , B f .akC1 /.! ı M /  EA t t t

and !,B,f .A/ 

n1 X kD0

We have

 EA

  

 ak akC1 1 f .ak /.! ı M /  EA , B . t t t

 

  ak akC1 ak akC1 , D EA , 1  EA ,1 . t t t t

It follows from Lemma 8.5.4 that        1 ak akC1 1 1 , B D ! .AB/.  .! ı M /  EA t t t ak akC1

262

Chapter 8 Heat Kernel Formulas and -function Residues

Hence,  n1 X kD0

  1 1 ! .AB/ !,B,f .A/ f .ak /  ak akC1

 n1 X kD0

  1 1 ! .AB/. f .akC1 /  ak akC1

Both coefficients in the latter formula converge to

Rb a

f .s/s 2 ds.

The proof of Theorem 8.5.1 can now be given. Proof of Theorem 8.5.1. Without loss of generality, f  0. For a given n 2 N, select fn 2 C 2 Œ0, 1/ supported on .1=2n, 2n such that 0 fn f and such that f D fn on .1=n, n. By assumption, jf .t /j const  minf1, t 2g, t > 0. It follows that 0 f  fn const  gn , where gn .t / :D minft 2 , n2 g C minf1, t 2n2 g,

t > 0.

We have 0 !,B,f .A/  !,B,fn .A/ const  kBk1 !,1,gn .A/. Since ! is a dilation invariant extended limit, it follows that   1 !,1,gn .A/ D z.! ı M / .minf.tA/2, n2 g/ t   1 2 2 C .! ı M / .minf1, n .tA/ g/ t     1 2 1 D .! ı M ı n /  min .tA/ , 2 t n     1 1 2 C .! ı M ı 1=n /  min 1, 2 .tA/ t n     1 2 1 . D .! ı M / .minf1, t 2g/ D O n t n Hence, 0 !,B,f .A/  !,B,fn .A/ O Therefore, !,B,fn .A/ ! !,B,f .A/ as n ! 1.

  1 . n

263

Section 8.6 -function Residues

Every function from C 2 Œ1=2n, 2n is of bounded variation on Œ1=2n, 2n. In particular, fn j.1=2n,2n is a difference of two monotone (on the interval .1=2n, 2n) functions. By Lemma 8.5.5, the equality (8.12) holds for fn . Hence, it also holds for the function f .

8.6 -function Residues We now shift our attention to -function residues, which are the functionals ! on M1,1 .M, /, ! an extended limit on L1 .0, 1/, introduced in the following definition. Observe that, when considering -function residues, we do not restrict our attention only to ! that are dilation invariant. We show in this section that -function residues provide fully symmetric functionals on M1,1 .M, / and can be identified with Dixmier traces for certain !. Definition 8.6.1. For every extended limit ! on L1 .0, 1/, the functionals ! : M1,1 .M, /C ! RC and !,B : M1,1 .M, /C ! C are defined by the formulas 

! .A/ :D !

 1 .A1C1=t / , t



!,B .A/ :D !

 1 .A1C1=t B/ t 0 A 2 M1,1 .M, /.

Here, B 2 M is an arbitrary operator. We begin by showing that the functionals given in Definition 8.6.1 are well defined on M1,1 .M, /C . Lemma 8.6.2. If ! : L1 .0, 1/ ! R is an extended limit, then ! .A/ < 1 and

!,B .A/ < 1 for any 0 A 2 M1,1 .M, /. Proof. It is clear that .s, A/  .1 C s/1 kAkM1,1 , s > 0. Therefore, Z .A1C1=t / kAk1C1=t M1,1

1 0

dt D t kAk1C1=t M1,1 . .1 C s/1C1=t

Hence, ! .A/ kAkM1,1 . It follows from j.A1C1=t B/j kBk1 .A1C1=t / that !,B .A/ kBk1 ! .A/. Lemma 8.6.3. For any 0 A, C 2 M1,1 .M, / we have .A1Cs C C 1Cs / ..A C C /1Cs / 2s .A1Cs C C 1Cs /,

s > 0.

264

Chapter 8 Heat Kernel Formulas and -function Residues

Proof. In the special case when M D L.H /, the first inequality can be found in [138, (2.9)]. In the general case, it follows directly from Proposition 4.6 (ii) of [92] when f .u/ D u1Cs , u > 0. The second inequality follows from the same proposition by setting there a D a D b D b  D 21=2 . The following theorem shows that the functionals ! are fully symmetric on M1,1 .M, /. Theorem 8.6.4. If ! : L1 .0, 1/ ! R is an extended limit, then ! extends to a fully symmetric linear functional on M1,1 .M, /. Proof. It follows from the left-hand side inequality of Lemma 8.6.3 that, for all positive operators A, C 2 M1,1 .M, /, we have

! .A C C /  ! .A/ C ! .C /. Noting that !.j21=t  1j/ D 0, it follows from the right-hand side inequality of Lemma 8.6.3 and Remark 6.3.5 that

! .A C C / ! .A/ C ! .C /. Therefore, we have

! .A C C / D ! .A/ C ! .C /. In other words, the functional ! is additive on M1,1 .M, /C . The homogeneity of

! follows from Remark 6.3.5. Thus, ! admits a unique extension to a linear functional on M1,1 .M, /. Finally, if C  0 and C  A 2 M1,1 .M, /C , then C , A 2 L1Cs .M, / and .C 1Cs / .A1Cs /. Hence, 1t .C 1C1=t / 1t .A1C1=t / and so ! .C / ! .A/. Hence, ! is a fully symmetric linear functional. The following is our main result concerning the -function residue. Theorem 8.6.5. If ! : L1 .0, 1/ ! R is an extended limit, then

!,B .A/ D ! .AB/,

0 A 2 M1,1 .M, /, B 2 M.

Proof. Suppose first that m B M for some strictly positive constants m, M . Applying Lemma A.2.9 in the appendix to the operators A and M 1 B (respectively, m1B), we have ms B 1=2 A1Cs B 1=2 .B 1=2 AB 1=2 /1Cs M s B 1=2 A1Cs B 1=2 .

265

Section 8.6 -function Residues

Therefore, 1 1 1 1=t m .A1C1=t B/ ..B 1=2 AB 1=2 /1C1=t / M 1=t .A1C1=t B/. t t t Since !.jm1=t  1j/ D 0 and !.jM 1=t  1j/ D 0, it follows from Remark 6.3.5 that

!,B .A/ D ! .B 1=2 AB 1=2 /. By Theorem 8.6.4, ! is a symmetric functional and, therefore, by Lemma 2.7.4 is a trace. It follows that ! .B 1=2 AB 1=2 / D ! .AB/. This proves the assertion of the theorem when B is bounded above and below by strictly positive constants. Let now 0 B 2 M be arbitrary and set Bn :D maxfB, 1=ng, n  1. From the first part of the proof, we have

!,Bn .A/ D ! .ABn /. Since ABn ! AB in M1,1 .M, / and since ! is a bounded functional (by Theorem 8.6.4), it follows that ! .ABn / ! ! .AB/ as n ! 1. On the other hand, j !,Bn .A/  !,B .A/j D j !,Bn B .A/j kBn  Bk1 ! .A/ ! 0,

n ! 1.

Therefore

!,B .A/ D lim !,Bn .A/ D lim ! .ABn / D ! .AB/. n!1

n!1

The assertion for arbitrary B 2 M follows by linearity. We now provide several formulas linking Dixmier traces and -function residues. Lemma 8.6.6. Let f : .0, 1/ ! R (respectively, ˇ : .0, 1/ ! R) be a positive decreasing (respectively, increasing) function. If ˇ.t / t for all t > 0, then Z 1 Z 1 f .t /dˇ.t / f .t /dt . 0

0

Proof. Without loss of generality, ˇ.0/ D 0. Let tk , 0 k n, n  1, be an increasing sequence such that t0 D 0. We have n1 X

f .tk /.ˇ.tkC1/  ˇ.tk // D

kD0

n1 X

ˇ.tkC1/.f .tk1 /  f .tk // C ˇ.tn /f .tn1 /

kD1

D

n1 X

kD1 n1 X

tkC1.f .tk1 /  f .tk // C tn f .tn1 /

f .tk /.tkC1  tk /.

kD0

266

Chapter 8 Heat Kernel Formulas and -function Residues

Hence, the Riemann–Stieltjes sums for the left-hand side do not exceed the Riemann– Stieltjes sum for the right-hand side. Theorem 8.6.7. Let ˇ : RC ! RC be an increasing function such that 0 ˇ.t / t , t  0. We have     Z 1 ˇ.t / h.t / e u=t dˇ.u/, t > 0 (8.16) D! , where h.t / :D ! t t 0 for any dilation invariant extended limit ! on L1 .0, 1/. Proof. First, note that Z

1

0

e 0

u=t

Z

1

dˇ.u/

e u=t du D t

0

by Lemma 8.6.6. Hence, the left-hand side of (8.16) is well defined. Since ! is dilation invariant, it follows that  Z 1   Z 1     1 h.t / 1 h.t / u=t n u=t n ! ! D .e / e dˇ.u/ D s ds  ! t 0 nC1 t t 0 for every n  0. Therefore,   Z 1    Z 1 h.t / 1 u=t u=t p.e /e dˇ.u/ D p.s/ds  ! ! t 0 t 0 for every polynomial p. For every f 2 C Œ0, 1, there exists a sequence pk , k  0, of polynomials such that pn ! f uniformly on Œ0, 1. Therefore,   Z 1    Z 1 h.t / 1 u=t u=t f .e /e dˇ.u/ D f .s/ds  ! ! t 0 t 0

(8.17)

for every f 2 C Œ0, 1. Set g.s/ :D s 1 Œe1 ,1 .s/, s 2 Œ0, 1. We have Z

1

g.e u=t /e u=t dˇ.u/ D ˇ.t /  ˇ.0/

0

and, therefore,     Z 1 ˇ.t / 1 u=t u=t g.e /e dˇ.u/ D ! . ! t 0 t

(8.18)

267

Section 8.6 -function Residues

Choose 0 f1k , f2k 2 C Œ0, 1 such that f1k # g and f2k " g almost everywhere. It follows from (8.17) and (8.18) that Z

1 0

       Z 1  h.t / h.t / ˇ.t / f2k .s/ds  ! f1k .s/ds  ! ! . t t t 0

Letting k ! 1, we obtain the assertion. With the above results on -function residues, we can connect -function residue functionals on M1,1 .M, / with Dixmier traces on M1,1 .M, /. Theorem 8.6.8. If ! is a dilation invariant extended limit ! on L1 .0, 1/ such that the extended limit ! ı log is still dilation invariant, then ! D !ılog. Proof. It is sufficient to verify the equality ! D !ılog on positive operators A 2 M1,1 .M, / such that A 1. Define a continuously increasing function ˇ : .0, 1/ ! .0, 1/ by setting Z 1 d nA ./ D .AEA .e u , 1//. ˇ.u/ :D  eu

Since nA .e u/ D O.ue u/ as u ! 1, it follows that Z cueu Z nA .eu / .s, A/ds .s, A/ds D O.u/. ˇ.u/ 0

0

Let h be as in Theorem 8.6.7 as applied to the above ˇ. We have Z 1 Z 1 u=t e dˇ.u/ D  e u.1C1=t /ud nA .e u/ D .A1C1=t /. h.t / D 0

0

Since ! ı log is dilation invariant, it follows from Definition 8.6.1 and Theorem 8.6.7 that     ˇ.t / h.t / D .! ı log/ . (8.19)

!ılog .A/ D .! ı log/ t t By Theorem 7.4.3, we have     Z 1 1 ˇ.t / . d nA ./ D .! ı log/ ! .A/ D ! log.t / 1=t t

(8.20)

The assertion follows immediately from (8.19) and (8.20) The next corollary follows by combining Theorems 8.6.5 and 8.6.8, we omit the proof.

268

Chapter 8 Heat Kernel Formulas and -function Residues

Corollary 8.6.9. If ! is a dilation invariant extended limit on L1 .0, 1/ such that the extended limit ! ı log is still dilation invariant, then   1 ! .AB/ D .! ı log/ .A1C1=t B/ , 0 A 2 M1,1 .M, /, B 2 M. t

8.7 Not Every Dixmier Trace is a -function Residue In this section we demonstrate that, unlike the situation with heat kernel functionals (see Theorem 8.3.6), the set of all -function residues is strictly smaller than the set of all normalized fully symmetric functionals. Hence, unlike heat kernels functionals,

-function residues do not represent all the Dixmier traces on M1,1 .M, /. Theorem 8.7.1. The set of Dixmier traces on M1,1 .M, / is strictly larger than the set of -function residues on M1,1 .M, /. The theorem is proved below, first we require two lemmas. Lemma 8.7.2. For the function x 2 C Œ0, 1, defined by the formula X nCu 2nCu2 , u 2 Œ0, 1, x.u/ :D n2Z

we have kxk1 < 2= log.2/. Proof. Evidently, sup x.u/

u2Œ0,1

X n2Z

sup 2nCu2

nCu

u2Œ0,1

D

X n2Z

sup

u2Œn,nC1

u

2u2 .

A direct computation yields

sup

u2Œn,nC1

2

u2u

8 n2n ˆ , ˆ sup! ! .A/ and the Dixmier trace !0 is not the residue of any

-function.

8.8 Notes -function Residues and Heat Kernel Functionals The interplay between Dixmier traces ! and the functionals  and ! on M1,1.M, / has been an important chapter in noncommutative geometry and has been treated (amongst many papers) in [9, 28, 32–34, 48, 212, 215]. We look at the history of some of these results. In [32], the equality   1 ! .AB/ D .! ı log/ .A1C1=t B/ D !ılog,B .A/, 0 A 2 M1,1.M, / (8.23) t was established for every B 2 M under strict conditions on !. These conditions were dilation invariance for both ! and !ılog and M -invariance of !. In [33], for the special case B D 1, the assumption that ! is M -invariant was removed. However, the case of an arbitrary B appears to be inaccessible by the methods in that article. In Section 8.6, we proved the general result which implies, in particular, that the equality (8.23) holds without requiring M -invariance of !. Our treatment followed the approach of [241]. In [32], the equality     1 1 ! .AB/ (8.24) ! .exp..tA/q /B/ D  1 C t q was established under the same strict conditions on ! and ! ı log as above. In [215], in the special case B D 1 the equality (8.24) was established under the assumption that ! is M invariant. However, again the case of an arbitrary B appears to be inaccessible by the methods in that article. In Section 8.5, we treated the case of an arbitrary operator B (again, mostly following [241]). Theorem 8.5.1 extends the results of [32,33] and gives an affirmative answer to the question stated in [9]. Previous articles [32, 33, 215, 241] stating similar results to the combination of Theorems 8.2.9 and 8.5.1 were based on the weak Karamata theorem (that is, Theorem 8.6.7) with the implicit condition ˇ.u/ u, u  0. This condition was not explicitly checked in [32, p. 92], [33, p. 276], [215, p. 570], including [241, p. 2466] where it can be seen that the condition nA .1=u/ D O.u/ (which is required to apply the weak Karamata theorem) in fact implies that the operator A 2 L1,1 .M, /. That is, in [215, 241] there is an error even though an alternative proof is given in [241, Theorem 49]. All these drawbacks are rectified in Theorem 8.5.1 and Corollary 8.5.2. Corollary 8.6.9 strengthens and extends the results of [33, Theorem 4.11] and [32, Theorem 3.8]. Various formulas of noncommutative geometry (in particular, those involving heat kernel estimates and generalized -functions) were established in [32,33,48] when the extended limit ! was assumed to be M -invariant. This class of extended limits was first introduced in [32] (see also [66]) and further studied and used in [6, 9, 33].

Chapter 9

Measurability in Lorentz Ideals

9.1 Introduction As we did in Chapter 8, we fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace . We explain the results of this chapter using the Dixmier–Macaev operator ideal M1,1 .M, /. The two main sets of Dixmier traces which we have considered in Chapter 7 and Chapter 8 are the set of all Dixmier traces ! (recall from Chapter 6 that this set coincides with the set of all normalized fully symmetric functionals on M1,1 .M, /) and the subset of Dixmier traces given by M -invariant extended limits, that is ! where ! D ! ı M is an extended limit on L1 .0, 1/. This subdivision in our consideration naturally leads to the following definition. Definition 9.1.1. An operator A 2 M1,1 .M, / is said to be (a) Dixmier measurable if the value ! .A/ is independent of the choice of the dilation invariant extended limit ! on L1 .0, 1/. (b) M -measurable if the value ! .A/ is independent of the choice of the M -invariant extended limit ! D ! ı M on L1 .0, 1/. The objective of this chapter is the characterization of the set of all positive Dixmier measurable operators and the set of all positive M -measurable operators in M1,1 .M, /. The main results are Theorems 9.2.1 and 9.3.1 and Theorem 9.5.1. Theorem 9.2.1 asserts that a positive operator A 2 M1,1 .M, / is Dixmier measurable if and only if there exists the limit 1 lim t !1 log.1 C t /

Z

t

.s, A/ds.

(9.1)

0

For a Dixmier measurable operator A, this limit calculates the value of every Dixmier trace of A. In Theorem 9.3.1, an equivalent condition is stated in terms of -function residues and heat kernel functionals. Theorem 9.5.1 provides the description of the positive M -measurable operators. More precisely, an operator 0 A 2 M1,1 .M, / is M -measurable if and only if  M

n

1 log.1 C t /

Z



t

.s, A/ds 0

! const,

n!1

273

Section 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals

in the Calkin algebra L1 .0, 1/=L01 .0, 1/. Using these descriptions Theorem 9.5.4 asserts that the two sets introduced in Definition 9.1.1 are distinct, that is, there exists 0 A 2 M1,1 .M, / which is M -measurable but not Dixmier-measurable. Knowing the Lidskii formula for a Dixmier trace (see Theorem 7.3.1) and the measurability result (9.1), it is natural to ask whether an arbitrary compact operator A in the Lorentz ideal M1,1 :D M1,1 .L.H /, Tr/ of compact operators is Dixmier measurable if and only if the limit n X 1 .j , A/ n!1 log.2 C n/

lim

j D0

exists, where .A/ is an eigenvalue sequence of A. Section 9.7 shows that this proposition is true if A 2 L1,1 where L1,1 :D L1,1 .L.H /, Tr/ is the weak-l1 ideal of compact operators. It also provides a counterexample that shows the result is not true for an arbitrary operator A 2 M1,1 . The notion of Dixmier measurability plays an important role in Part IV, and its use will be observed in subsequent chapters.

9.2 Positive Dixmier Measurable Operators in Lorentz Ideals For this chapter, fix an atomless or atomic von Neumann algebra M equipped with a faithful normal semifinite trace , and that the symbol M : L1 .0, 1/ ! L1 .0, 1/ denotes the logarithmic mean 1 .M x/.t / :D log.t /

Z

t

x.s/ 1

ds , s

x 2 L1 .0, 1/, t > 0.

In this section, we classify positive Dixmier measurable operators for the Lorentz operator ideal M .M, / associated to a Lorentz function space M (see Example 2.6.10). The main result of this section is the next theorem.

Theorem 9.2.1. Let fies the condition

: RC ! RC be a concave and increasing function that satislim

t !1

.2t / D 1. .t /

(9.2)

For a positive operator A 2 M .M, / the following conditions are equivalent. (a) A is Dixmier measurable.

274

Chapter 9 Measurability in Lorentz Ideals

(b) There exists the limit lim

t !1

1 .t /

Z

t

.s, A/ds. 0

It is clear from the theorem that, if 0 A 2 M .M, / is Dixmier measurable, then Z t 1 .s, A/ds ! .A/ D lim t !1 .t / 0 for every Dixmier trace ! on M .M, / (that Dixmier traces exist when satisfies the condition (9.2) is given by the results in Chapter 6). To prove the theorem we use the following lemmas. Lemma 9.2.2. If

satisfies the condition (9.2), then, for every T > 0, we have lim

t !1

.T t / D 1. .t /

Proof. Without loss of generality, T > 1. Select n 2 N such that T 2n . We have 1

.T t / .t /

n1 Y .2nt / D .t /

kD0

.2kC1t / . .2k t /

Evidently, lim

t !1

.2kC1t / D 1, .2k t /

k  0.

Therefore, 1 lim inf t !1

.T t / lim sup t !1 .t /

.T t / 1. .t /

The assertion follows immediately. Lemma 9.2.3. If satisfies the condition (9.2), then, for every increasing function  , we have     .t / .t /   D lim sup ,   D lim inf .  t !1 t !1 .t / .t / Here,  : L1 .0, 1/ ! R is given by the formula (6.3). Proof. We prove the first assertion only. The proof of the second one is identical. Denote the right-hand side of the first formula by C .

Section 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals

275

Due to Lemma 9.2.2, for every T > 1, we can select N0 .T / sufficiently large such that   1 1 .N T / 1 C .N /, N > N0 .T /. T Select N > N0 .T / such that   1 .N / C 1 . .N / T For every s 2 ŒN , N T , we have       .N / .N / 1 1 1  C 1  1C DC 1 2 . .N T / .N / T T T

.s/ .N /  D .s/ .N T /

In particular, for every T > 1 (and for selected N ), we have Z

NT N

  .s/ ds 1  C 1  2 log.T /. .s/ s T

The assertion now follows from the definition of . With these results we can prove Theorem 9.2.1. Proof of Theorem 9.2.1. Set Z

t

.s, A/ds,

.t / :D

t > 0.

0

By the Hahn–Banach theorem, there exist functionals !1 , !2 2 L1 .0, 1/ such that             !1 , !2 , !1 D , !2 D   . By Lemma 6.2.7, the functionals !1 and !2 are dilation invariant extended limits. If A is Dixmier measurable, it follows from Lemma 9.2.3 that         .t /    D   D !2 D !2 .A/ D !1 .A/ D !1 lim inf t !1 .t /   .t /  D lim sup . D t !1 .t /

276

Chapter 9 Measurability in Lorentz Ideals

9.3 Positive Dixmier Measurable Operators in M1,1 In this section we characterize Dixmier measurable operators in the Dixmier–Macaev operator ideal M1,1 .M, /. The function log.1 C t /, t  0, satisfies condition (9.2), and therefore the positive Dixmier measurable operators in the Dixmier–Macaev operator ideal satisfy Theorem 9.2.1. As seen in Chapter 8, Dixmier traces on the Dixmier– Macaev operator ideal can be represented by heat kernel functionals and -function residues. Equivalent characterizations of Dixmier measurability, in terms of heat kernel functionals and -function residues are given below. The following theorem is the main result of this section. Theorem 9.3.1. Let A 2 M1,1 .M, / be a positive operator. The following conditions are equivalent. (a) The operator A is Dixmier measurable. (b) There exists the limit 1 t !1 log.1 C t /

Z

t

.s, A/ds.

lim

(c) There exists the limit

0



 1 .A/1 .e / . lim M  !1

(d) There exists the limit lim s.A1Cs /.

s!0

Furthermore, if any of the conditions (a)–(d) above holds, then we have the coincidence of the three limits   Z t 1 1 1 .e .A/ / D lim s.A1Cs / .s, A/ds D lim M lim t !1 log.1 C t / 0 s!0  !1 with the value of a Dixmier trace of A. The theorem is proved further below. We first discuss the presence of the logarithmic mean M in Theorem 9.3.1. The following corollary follows immediately from Theorem 9.3.1. Corollary 9.3.2. Let A 2 L1,1 .M, / be a positive operator. If there exists the limit 1 1 .e .A/ /, !1  lim

Section 9.3 Positive Dixmier Measurable Operators in M1,1

277

then there also exists the limit 1 lim t !1 log.1 C t /

Z

t

.s, A/ds. 0

The following example shows that we cannot omit the logarithmic mean M from the statement of Theorem 9.3.1 (c). Example 9.3.3. There exists a positive operator A 2 L1,1 .M, / such that 1 lim t !1 log.1 C t /

Z

t

.s, A/ds D 0

0

and 1 1 lim sup .e .A/ / > 0. !1  Proof. Define a positive operator A by setting 8 1 ˆ e, let n D Œlog.log.t //. It follows that Z

t

Z

ee

.s, A/ds

nC1

.s, A/ds D O.n log.n// D o.log.t //.

0

0

On the other hand, we have en 1 Z 1  X 1 X ne 1 1 1 en 1 en .A/1 .e e  e  e n e . / e  s ds D n   ee

nD0

nD0

278

Chapter 9 Measurability in Lorentz Ideals n

For a given n 2 N, set  D e e . It follows that 1 1 1 en 1 en .e .A/ /  e  e  e n e D e 1  e n.  Therefore, 1 1 lim sup .e .A/ /  e 1 . !1  To prove Theorem 9.3.1 we start with a well-known Tauberian result. The proof can be found in Hardy [108, Theorem 64]. For this section, the symbol C denotes the Cesàro operator C : L1 .0, 1/ ! L1 .0, 1/ given by 1 .C x/.t / :D t

Z

t

x.s/ds,

x 2 L1 .0, 1/, t > 0.

0

Lemma 9.3.4. Let z 2 L1 .0, 1/ be a positive differentiable function and let a 2 R. If t z 0 .t / is bounded from below and if .C z/.t / ! a as t ! 1, then z.t / ! a as t ! 1. Our next lemma plays an important role in the proof of Theorem 9.3.1. Lemma 9.3.5. Let z be a positive locally integrable function on .0, 1/ and let a 2 R. If M z 2 L1 .0, 1/ and if .M 2 z/.t / ! a as t ! 1, then .M z/.t / ! a as t ! 1. Proof. Set x :D .M z/ ı exp 2 L1 .0, 1/. We claim that x satisfies the conditions of Lemma 9.3.4. We have 1 .M z/.t / D log.t /

Z

2

1

t

1 du D .M z/.u/ u log.t /

Z

log.t /

x.s/ds, 0

where we have used the substitution u D e s in the second equality. By the conditions on z, we have .C x/.log.t // ! a as t ! 1. Equivalently, .C x/.t / ! a as t ! 1. The condition remaining to be shown is that tx 0 .t / is bounded from below. We have   Z et Z t 1 ds 0 1 e ds tx .t / D t C z.e t /. z.s/ D z.s/ t 1 s t 1 s 0

Since z is positive, we have tx 0.t /  .M z/.e t / and since M z 2 L1 .0, 1/, we conclude that tx 0 .t /  const. By Lemma 9.3.4, we have x.t / ! a as t ! 1 and, therefore, .M z/.t / ! a as t ! 1.

Section 9.3 Positive Dixmier Measurable Operators in M1,1

279

Lemma 9.3.6. Let x 2 L1 .0, 1/ and let a 2 R. The following conditions are equivalent. (a) lim inf x.t / a lim sup x.t /. t !1

t !1

(b) There exists an extended limit on L1 .0, 1/ such that .x/ D a. Proof. The implication (b) ) (a) follows immediately from the definition of an extended limit. In order to prove the implication (a) ) (b), define a functional on R C xR by setting .˛ C ˇx/ :D ˛ C ˇa. Clearly, .z/ lim sup z.t /, t !1

z 2 R C xR.

The assertion now follows from the Hahn–Banach theorem. The following is the classical Karamata theorem. The proof is identical to that of Theorem 8.6.7 and hence it is omitted. Theorem 9.3.7. Let ˇ be a continuous increasing function. Set Z 1 q h.t / D e .u=t / dˇ.u/. 0

We have

  1 h.t / ˇ.t / D 1C lim t !1 t q t !1 t lim

provided that the left-hand side limit exists. Here  is the Gamma function and q > 0. We can now prove Theorem 9.3.1. Proof of Theorem 9.3.1. The implication (b) ) (a) follows from the definition of ! . (a) ) (b). Suppose that ! .A/ D a for every dilation invariant extended limit !. It follows from Lemma 7.4.1 that ıM .A/ D a for every extended limit . That is, we have the equality   Z t 1 .s, A/ds D a. . ı M / log.1 C t / 0 It follows from Lemma 9.3.6 that   Z t 1 lim M .s, A/ds D a. t !1 log.1 C t / 0

(9.3)

280

Chapter 9 Measurability in Lorentz Ideals

Set z.t / :D t.t , A/. Observe that z is a positive measurable, but not necessarily bounded, function. Note, however, that Z t 1 .M z/.t / D .s, A/ds. log.t / 1 Since A 2 M1,1 .M, / it follows that M z 2 L1 .0, 1/. Observe that Z t 1 .s, A/ds D 0. lim .M z/.t /  t !1 log.1 C t / 0

(9.4)

Recall that .My/.t / ! 0 as t ! 1 whenever y 2 L1 .0, 1/ is such that y.t / ! 0 as t ! 1. Combining (9.3) and (9.4), we infer that .M 2 z/.t / ! a as t ! 1. By Lemma 9.3.5, we infer that .M z/.t / ! a as t ! 1. The proof of the implication is completed by referring to (9.4). (c) ) (a). Let a be the limit in (c). By definition of the functional ! , we have that ! .A/ D a for every dilation invariant extended limit !. By Theorems 6.4.1 and 8.3.6, the set of all Dixmier traces coincides with that of all heat kernel functionals. Hence, ! .A/ D a for every dilation invariant extended limit !. (a) ) (c). Suppose that ! .A/ D a for every dilation invariant extended limit !. The same argument as above shows that ! .A/ D a for every dilation invariant extended limit !. It follows from Lemma 7.4.1 that ıM .A/ D a for every extended limit . That is,   1 2 .A/1 . ı M / .e / D a.  It follows from Lemma 9.3.6 that lim M

t !1

 2

 1 .A/1 .e / D a. 

We know that the mapping  1 .A/1 .e / !M  

is bounded. The proof of the implication is completed by using Lemma 9.3.5. (a) ) (d). Suppose that ! .A/ D a for every dilation invariant extended limit !. It follows from Theorems 8.6.4 and 6.4.1 that the set of all -function residues is a subset of the set of Dixmier traces. Hence, for every extended limit , we have   1 1C1=t .A / D a. t Applying Lemma 9.3.6 completes the proof of the implication.

281

Section 9.4 C -invariant Extended Limits

(d) ) (a). Without loss of generality, kAk1 1. By assumption, we have lim

r !1

1 .A1C1=r / D a. r

Define a continuous increasing function ˇ on .0, 1/ by setting Z u e v d nA .e v /. ˇ.u/ :D  0

It is clear that Z 1 Z 1 Z 1 u=r u.1C1=r / u e dˇ.u/ D  e d nA .e / D  1C1=r d nA ./ h.r / :D 0 0 0 Z 1 D 1C1=r d nA ./ D .A1C1=r /. 0

It follows from Theorem 9.3.7 that a D lim

r !1

h.r / ˇ.u/ D lim . u!1 u r

Therefore, ˇ.u/ 1 D lim u!1 u u!1 u

a D lim

Z

1

eu

1 t !1 log.1 C t /

Z

1

d nA ./ D lim

Hence, 1 lim t !1 log.1 C t /

Z

d nA ./. 1=t

1 log.t /=t

d nA ./ D a.

The assertion now follows from Theorem 7.2.1.

9.4 C -invariant Extended Limits Define the Cesàro operator C : L1 .R/ ! L1 .R/ by Z 1 t .C x/.t / :D x.s/ds, x 2 L1 .R/, t 2 R. t 0 This is a slight extension of the operator C from the last section (and Section 4.5) in that t can take negative values (and the order of integration is reversed in that case so Rt R0 that 0 :D  t , t < 0). The main objective of this section is to characterize the range !.x/ where x 2 L1 .R/ is real-valued and fixed and ! varies over all C -invariant extended limits on

282

Chapter 9 Measurability in Lorentz Ideals

L1 .R/. By an extended limit on L1 .R/ we mean a state on the algebra L1 .R/ that vanishes on functions that have support in an interval .1, t , for some t 2 R. The main result is given in Theorem 9.4.3 and a simpler criterion is given in Theorem 9.4.9. The latter result is used in Theorem 9.5.1 in the next section to present a description of the set of all positive M -measurable operators. Lemma 9.4.1. For every real-valued x 2 L1 .R/ we have lim sup.C x/.t / lim sup x.t /. t !C1

t !C1

Proof. It is sufficient to prove the assertion for x C kxk1 instead of x. Therefore, we may assume that x  0. For every fixed t0 > 0 we have lim sup.C x/.t / D lim sup.C x .t0,1/ /.t / sup .C x/.t / sup x.t /. t !C1

t t0

t !C1

t t0

If t0 ! C1, the right-hand side converges to lim supt !C1 x.t /. Lemma 9.4.2. For every real-valued x 2 L1 .R/, the following limit exists 1 lim sup p.x/ :D lim n!1 n t !C1

 n1 X

 C x .t /. k

kD0

So defined, the functional p is convex and homogeneous. Proof. Set ˛n :D lim sup t !C1

 n1 X

 C x .t /, k

n 2 N.

kD0

We have ˛nCm D lim sup

 n1 X

t !C1

lim sup t !C1

  m1  X C k x .t / C C n C k x .t /

kD0

X n1 kD0

kD0

   m1 X k n k C x .t / C lim sup C C x .t /. t !C1

It follows from Lemma 9.4.1 that ˛nCm ˛n C ˛m . The existence of the limit follows from Lemma 3.4.6.

kD0

(9.5)

283

Section 9.4 C -invariant Extended Limits

The next theorem is the main technical result of this section. Theorem 9.4.3. Let x 2 L1 .R/ be real-valued and let a 2 R. The following conditions are equivalent.    n1  n1 X X 1 1 k k (a) lim lim inf lim sup C x .t / a lim C x .t /. n!1 n t !C1 n!1 n t !C1 kD0

kD0

(b) There exists an extended limit ! on L1 .R/ such that ! D ! ı C and !.x/ D a. Proof. (a) ) (b). Let p be the functional defined in Lemma 9.4.2. We have p.x/ a p.x/. Define the functional ! on R C xR by the formula !.1 C 2 x/ D 1 C 2 a,

1, 2 2 R.

We have that ! p on R C xR. Indeed, for 1 , 2 2 R, !.1 C 2 x/ D 1 C 2a 1 C j2 jp.sgn.2 /x/ D p.1 C 2x/. By the Hahn–Banach theorem, the functional ! can be extended to L1 .R/ preserving the inequality ! p. If z  0 then .p.z// !.z/ p.z/. It follows that ! is a positive functional since !.z/  p.z/  0. It also follows that !.1/ D 1 since 1 D p.1/ !.1/ p.1/ D 1. Hence, ! is a state. Suppose that z.t / ! 0 as t ! 1. It follows that, for every k 2 N, .C k z/.t / ! 0 as t ! 1. Hence, for every n 2 N, we have lim

 n1 X

t !1

 C k z .t / D 0

kD0

and, therefore, p.z/ D p.z/ D 0. Thus, !.z/ D 0. It follows that ! is an extended limit. Note that n1 X C k .1  C / D 1  C n . kD0

Therefore, p.z  C z/ D lim

n!1

1 lim sup.z  C n z/ D 0. n t !1

Similarly, p.C z z/ D 0. Therefore, !.z C z/ D 0 for every z 2 L1 .R/. It follows that ! D ! ı C .

284

Chapter 9 Measurability in Lorentz Ideals

(b) ) (a). Let x 2 L1 .R/ be real-valued and let ! be a C -invariant extended limit. We have  n1  n1   X 1X k 1 k !.x/ D ! C x lim sup C x .t /. n n t !1 kD0

kD0

Taking the limit n ! 1, we conclude that !.x/ p.x/. It follows that p.x/ !.x/ p.x/. Theorem 9.4.3 can be used to describe the positive M -measurable operators. However, the following lemmas lead to a much simpler description. Lemma 9.4.4. For every real-valued x 2 L1 .R/, the following limit exists q.x/ D lim lim sup.C n x/.t /. n!1 t !1

Proof. The assertion follows from Lemma 9.4.1. Lemma 9.4.5. For every real-valued x 2 L1 .R/, we have p.x/ q.x/. Proof. For n  0 set ˛n :D lim sup.C n x/.t /. t !1

Evidently, lim sup t !1

 n1 X kD0

 n1 X C k x .t / ˛k ,

n  0.

kD0

Therefore, n1 1X ˛k  p.x/. q.x/ D lim ˛n D lim n!1 n!1 n kD0

The proof of the converse inequality is more complicated. Lemma 9.4.6. Let 0 x 2 L1 .R/. For every  > 0, there exists n 2 N and (an arbitrary large) t0 > 0 such that .C n x/.t /  .1  /q.x/ for every t 2 Œt0 , t0=4. Proof. Without loss of generality, q.x/ > 0. Fix 0 <  < 1 and choose n 2 N such that   1 2 n1 x/.t / q.x/ 1 C  . lim sup.C t !1 2

285

Section 9.4 C -invariant Extended Limits

Hence, there exists t1 > 0 such that .C n1 x/.t / q.x/.1 C  2 / for every t  t1 . If   1 n lim inf.C x/.t /  1   q.x/, t !1 2 then the assertion is evident. Suppose then that   1 n lim inf.C x/.t / < 1   q.x/, t !1 2 and note that lim sup.C n x/.t /  q.x/. t !1

Select (an arbitrarily large) t2 such that .C n x/.t2/ D .1   2 /q.x/. Let     1 n t0 D sup t 2 Œt1 , t2 : .C x/.t / D 1   q.x/ . 2 It follows that Z

t2

.1   /t2q.x/ D 2

.C

n1

0

Z

t0

x/.s/ds D

.C

n1

Z x/.s/ds C

0

t2

.C n1 x/.s/ds

t0

t0.C n x/.t0 / C .t2  t0/.1 C  2/q.x/    D t0 1  q.x/ C .t2  t0 /.1 C  2 /q.x/. 2 Hence,

  1 .1   /t2 t0 1   C .t2  t0/.1 C  2/. 2 2

Simple arithmetic gives us that t0 4t2. The assertion now follows from the definition of t0 . Lemma 9.4.7. Let 0 x 2 L1 .R/. For every natural number r  2, there exist m 2 N and (an arbitrary large) t 2 R such that   2 .C mCs x/.t /  1  q.x/, 0 s < r , (9.6) r where s is a natural number. Proof. Due to Lemma 9.4.6, there exists m 2 N and (an arbitrary large) n 2 N such that   1 m t 2 Œn, r 2r n. (9.7) .C x/.t /  q.x/ 1  2 r

286

Chapter 9 Measurability in Lorentz Ideals

We claim that, for every natural number 0 s < r , we have   1 sC1 q.x/, .C mCs x/.t /  1  2 r

t 2 Œr 2s n, r 2r n.

(9.8)

The inequality (9.8) holds for s D 0 because of (9.7). We use an argument by induction. Suppose that (9.8) holds for some 0 s < r  1. It follows that, for every t 2 Œr 2.sC1/n, r 2r n, .C

mCsC1

Z 1 t x.s/ds  .C mCs x/.s/ds 2s t 0 nr     1 sC1 t  nr 2s 1 sC2  1 2  q.x/  1  2 q.x/. t r r

1 x/.t / D t

Z

t

Hence (9.8) is shown for s C 1, and the claim is shown. It follows from (9.7) that, for t D r 2r n and 0 s < r ,     1 r 2 mCs x/.t /  1  2 q.x/  1  .C q.x/. r r Lemma 9.4.8. For every real-valued x 2 L1 .R/, we have p.x/  q.x/. Proof. It is sufficient to prove the assertion for x C kxk1 instead of x. Therefore, without loss of generality, we assume that x  0. Fix r 2 N. By Lemma 9.4.7, there exists m 2 N and an arbitrary large t such that X r 1

 C mCk x .t /  .r  2/q.x/.

kD0

Therefore, lim sup t !1

X r 1

 C mCk x .t /  .r  2/q.x/.

(9.9)

kD0

It follows from (9.9) and Lemma 9.4.1 that 1 lim sup r t !1

X r 1 kD0

   r 1 X 1 2 q.x/. C k x .t /  lim sup .C mCk x/.t /  1  r t !1 r kD0

Taking the limit r ! 1, the assertion follows. Theorem 9.4.9. Let x 2 L1 .R/ be real-valued and let a 2 R. The following conditions are equivalent.

287

Section 9.5 Positive M -measurable Operators

(a)

lim lim inf.C n x/.t / a lim lim sup.C n x/.t /.

n!1 t !1

n!1 t !1

(b) There exists an extended limit ! on L1 .R/ such that ! D ! ı C and !.x/ D a. Proof. The assertion follows from Theorem 9.4.3, Lemma 9.4.5 and Lemma 9.4.8. Let L01 .0, 1/  L1 .0, 1/ denote the subspace of (equivalence classes of) bounded functions that vanish at infinity. Corollary 9.4.10. Let x 2 L1 .0, 1/ be real-valued and let a 2 R such that !.x/ D a for every extended limit ! on L1 .R/ such that ! D ! ıC . It follows that C n x ! a in the Calkin algebra L1 .0, 1/=L01 .0, 1/. Proof. It follows from Theorem 9.4.9 that lim lim sup.C n x/.t / D a.

n!1 t !1

Hence, .C n x  a/C ! 0 in the Calkin algebra L1 .0, 1/=L01 .0, 1/. Similarly, .C n x  a/ ! 0 in the Calkin algebra. Therefore, C n x  a D .C n x  a/C  .C n x  a/ ! 0 in the Calkin algebra.

9.5 Positive M -measurable Operators We are now able to characterize the positive M -measurable operators in the Dixmier– Macaev operator ideal M1,1 .M, /. The characterization given in Theorem 9.5.1 below, and the characterization of Dixmier measurable operators given in Theorem 9.3.1, allows us to show that there is a positive M -measurable operator which is not Dixmier measurable (Theorem 9.5.4). Our technique is to transfer the results about C -invariant extended limits on L1 .R/ from the last section to M -invariant extended limits on L1 .0, 1/. We do this using the exponential map exp : R ! .0, 1/ or, equivalently, exp : L1 .0, 1/ ! L1 .R/ where exp.x/ :D x ı exp, x 2 L1 .0, 1/. For every t > 0, and x 2 L1 .0, 1/, we have ..M x/ ı exp/.t / D

1 t

Z

et

x.s/ 1

1 ds D s t

Z

t

x.e u/du D C.x ı exp/.t /.

0

Rb Ra When t < 0 the same identity equality holds, understanding that a :D  b if b < a. We conclude that we have that .M x/ıexp D C.x ıexp/ for all x 2 L1 .0, 1/. Thus,

288

Chapter 9 Measurability in Lorentz Ideals

an extended limit ! on L1 .R/ is C -invariant if and only if the extended limit ! ı exp on L1 .0, 1/ is M -invariant, since .! ı exp/.M x/ D .! ı C /.exp.x//,

x 2 L1 .0, 1/.

In this section, L01 .0, 1/  L1 .0, 1/ is the subspace of (equivalence classes of) bounded functions that vanish at infinity. Theorem 9.5.1. A positive operator A 2 M1,1 .M, / is M -measurable if and only if   Z t 1 n .s, A/ds ! const M log.1 C t / 0 in the Calkin algebra L1 .0, 1/=L01 .0, 1/. Proof. An extended limit ! on L1 .R/ is C -invariant if and only if the extended limit ! ı exp on L1 .0, 1/ is M -invariant. Hence, A is M -measurable if and only if there exists a  0 such that  !

1 log.1 C e t /

Z



et

D a,

.s, A/ds

! D ! ı C.

0

It follows from Corollary 9.4.10 that  C

n

1 log.1 C e t /

Z



et

.s, A/ds

!a

0

in the Calkin algebra L1 .0, 1/=L01 .0, 1/. The assertion follows immediately. We use the characterization in Theorem 9.5.1 to show that there is an M -measurable positive operator in M1,1 .M, / which is not Dixmier measurable. Consider the following operator K : L2 .1, e/ ! L2 .1, e/ 1 .Kx/.t / :D t

Z

t 1

1 x.s/ds C t .e  1/

Z

e

x.s/ds,

x 2 L2 .1, e/.

1

In the rest of this section we use the notation xn  x to denote uniform convergence in C Œ1, e. Lemma 9.5.2. For every x 2 C Œ1, e, we have Z e ds K nx  x.s/ . s 1

289

Section 9.5 Positive M -measurable Operators

Proof. The operator K : L2 .1, e/ ! L2.1, e/ is Hilbert–Schmidt and is, therefore, compact. Hence, the (non-zero part of the) spectrum .K/ of K consists of discrete eigenvalues with finite algebraic mutliplicity. We claim that .K/ D f0g [ f1g and 1 is a simple eigenvalue. Indeed, let  ¤ 0 be an eigenvalue and let x 2 L2 .1, e/ be the corresponding eigenfunction. We have 1 t

Z

t 1

1 x.s/ds C t .e  1/

Z

e

x.s/ds D x.t /.

(9.10)

1

Since x 2 L1 .1, e/, the left-hand side depends continuously on t . Thus, x 2 C Œ1, e. Also the left-hand side is a continuously differentiable function of t and, therefore, so is x. Multiplying both sides of (9.10) by t and taking derivatives, we obtain that x.t / D .tx.t //0. Therefore, x.t / D C t .1/= . Substituting into (9.10), we obtain  D 1 and, therefore, x D 1. This proves the claim. Let z.t / D t 1 for t 2 Œ1, e and let L be the hyperplane in L2.1, e/ orthogonal to z. Note that K  z D z. Therefore, .Kx, z/ D .x, K  z/ D .x, z/ D 0,

x 2 L.

Hence, K : L ! L. Since 1 … L, it follows that .K : L ! L/ D f0g and, therefore, K : L ! L is a quasi-nilpotent operator. Hence, K n x ! 0 in L2 .1, e/ for every x 2 L. Evidently, kxk1 D K.kxk1 / Kx K.kxk1/ D kxk1 . Therefore, K : C Œ1, e ! C Œ1, e is a contraction. Observe that k.Kx/0 k1 .e C 1/kxk1 for every x 2 C Œ1, e. Hence, the image of the unit ball under K is an equicontinuous set. It follows from the Arzela–Ascoli theorem that K : C Œ1, e ! C Œ1, e is a compact operator. Let x 2 C Œ1, e \ L and let z be a limit point of the sequence K nx, n  0. Since n K x ! 0 in L2.1, e/, it follows that z D 0. Thus, K nx ! 0 in C Œ1, e. For every x 2 C Œ1, e, we have x  .x, z/ 2 C Œ1, e \ L. Therefore, K n .x  .x, z//  0. Since K1 D 1, the assertion follows. Define the mapping  : .0, 1/ ! Œ1, e by setting .t / :D t e Œlog.t / ,

t > 0.

(9.11)

290

Chapter 9 Measurability in Lorentz Ideals

Lemma 9.5.3. For every x 2 C Œ1, e, we have Z e ds n x.s/ . C .x ı /  s 1 Proof. Fix t > 0 and set k :D Œlog.t /. It is clear that   Z t Z iC1 1 X e x.e i s/ds C x.e k s/ds t ek ei i 1, we have Z s du 1 x.u/ ds log.s/ u 0 1 1  Z 1 Z t Z t x.u/ 1 ds 1 .M x/.s/ds C du. D t 0 t 1 u u log.s/

.CM x/.t / D

However,

1 t

Z

1

.M x/.s/ds C

Z

t u

1 t

Z

t

ds D li.t /  li.u/, log s

293

Section 9.6 Additional Invariance of Dixmier Traces

where li is the logarithmic integral function defined by the formula Z

t e

ds . log.s/

Z

t

li.t / :D Therefore, 1 .CM x/.t / D t 1 D t

Z

1

0

Z

0

1

li.t / .M x/.s/ds C t

1

du 1 x.u/  u t

Z

t

x.u/ 1

1 li.t / log.t /  .M x/.t /  .M x/.s/ds C t t

Z

li.u/ du u

t

x.u/ 1

li.u/ du. u

For the function li, we have the following asymptotic formula li.t / D

t .1 C o.1//. log.t /

(9.13)

Therefore, for every t > e, we have ˇ ˇZ t Z t ˇ li.u/du ˇˇ jli.u/jdu ˇ x.u/ kxk D kxk1 .log.t /  li.t /  t C 2e  1/. 1 ˇ ˇ u u 1 1 It now follows from (9.13) that 1 t !1 t

Z

t

x.u/

lim

1

li.u/ du D 0. u

Therefore, .CM x/ 2 .1 C o.1//  M x C L01 .0, 1/ for every x 2 L1 .0, 1/. It follows that CM  M maps L1 .0, 1/ into L01 .0, 1/. An important corollary of Lemma 9.6.1 is: Corollary 9.6.3. Every M -invariant extended limit on L1 .0, 1/ is also C -invariant. Proof. Let ! be an M -invariant extended limit and let x 2 L1 .0, 1/. It follows from Lemma 9.6.1 that !.C x/ D !.M C x/ D !.M x/ D !.x/. Lemma 9.6.4. For every x 2 L1 .0, 1/, the function .C x/ ı exp is uniformly continuous.

294

Chapter 9 Measurability in Lorentz Ideals

Proof. It is clear that

 Z ..C x/.e // D e t

0

et

0

t

x.s/ds

D e t

Z

0

et

x.s/ds C x.e t /.

0

Therefore, ..C x/ ı exp/0 D .C x/ ı exp Cx ı exp 2 L1 .0, 1/. Hence, .C x/ ı exp is a Lipschitz function and, therefore, is uniformly continuous. Lemma 9.6.5. Let s ! hs be a norm-continuous map from the interval Œa, b into L1 .0, 1/. If ! 2 L1 .0, 1/ , then the function s ! !.hs / is Riemann-integrable and Z b  Z b ! hs ds D !.hs /ds. a

a

Proof. Fix  > 0 and select n 2 N such that khs1  hs2 k1 ,

8s1 , s2 > 0 : js1  s2 j

ba . n

It follows that Z

b a

n1 ba X hs ds  haCi .ba/=n .b  a/. n 1 i D0

Thus, ba X haCi .ba/=n  n n1

Z

i D0

b a

hs ds

as n ! 1. It follows that ba X !.haCi .ba/=n / ! ! n n1 i D0

Z

b a

 hs ds .

Evidently, j!.hs1 /  !.hs2 /j k!kL1 khs1  hs2 k1 . Hence, the mapping s ! !.hs / is continuous and, therefore, Riemann-integrable. Hence, Z b n1 ba X !.haCi .ba/=n / ! !.hs /ds. n a i D0

The assertion follows immediately.

295

Section 9.6 Additional Invariance of Dixmier Traces

Lemma 9.6.6. Let ! be a dilation invariant extended limit on L1 .0, 1/. For every x 2 L1 .0, 1/ such that x ı exp is uniformly continuous, we have !.x/ lim sup sup.M1=t x/.s/. s!1 t >0

Proof. Fix s > 0. Since x ı exp is uniformly continuous, it follows that the mapping u ! 1=ux is norm-continuous on the interval Œ1, s. Therefore,  Z s Z s du du D D !.x/  log.s/. 1=u x !.1=u x/ ! u u 1 1 Note that .1=u x/.t / D .1=t x/.u/. Therefore,   Z s Z s du du 1 1 1=u x .1=t x/.u/ sup !.x/ D ! log.s/ 1 u t >0 log.s/ 1 u D sup.M1=t x/.s/. t >0

Taking the limit s ! 1, we obtain the assertion. Lemma 9.6.7. Let x 2 L1 .0, 1/ be such that x ı exp is uniformly continuous. For every dilation invariant extended limit ! on L1 .0, 1/ and for every n 2 N, we have !.x/ D !.C n x/. Proof. By Lemma 9.6.6, we have !.x  C x/ lim sup sup..M1=t /.x  C x//.s/. s!1 t >0

However, 1=t commutes with C . Therefore, !.x  C x/ lim sup sup..M.1  C //.1=t x//.s/. s!1 t >0

It follows from (9.12) that ..M C  M /z/.s/ D

.C z/.1/  .C z/.s/ . log.s/

Hence, j..M C  M /z/.s/j

2kzk1 . j log.s/j

Substituting z D 1=t x, we obtain !.x  C x/ lim sup s!1

2kxk1 D 0. j log.s/j

Substituting x ! x, we obtain !.C x  x/ 0. Therefore, !.x/ D !.C x/.

296

Chapter 9 Measurability in Lorentz Ideals

By Lemma 9.6.4, .C k x/ ı exp is uniformly continuous for every k 2 N. Note that x  C n x D .1  C /z,

zD

n1 X

C k x.

kD0

Applying the result of the previous paragraph to z, we obtain !.x  C n x/ D !.z  C z/ D 0. The assertion follows immediately. Lemma 9.6.8. For every dilation invariant extended limit ! on L1 .0, 1/, there exists a dilation and C -invariant extended limit !0 on L1 .0, 1/ such that !.x/ D !0 .x/ whenever x 2 L1 .0, 1/ is such that x ı exp is uniformly continuous. Proof. Let ! be a dilation invariant extended limit and let E be the set of all x 2 L1 .0, 1/ such that x ı exp is uniformly continuous. We apply Theorem 6.2.5 to the commutative semigroup G D fC n1=t : n  0, t > 0g, the linear subset E  L1 .0, 1/, the linear functional !jE and the convex functional lim sup . The restriction of ! on E can be extended to the dilation and C -invariant functional !0 2 L1 .0, 1/ . Since !0 .z/ lim sup z.t /, t !1

z 2 L1 .0, 1/,

it follows that !0 is an extended limit. Recall that ! D !0 on E by the definition of !0 . The assertion is proved. The following theorem proves that, in the construction of a Dixmier trace ! on M1,1 .M, / given in Section 6.3 (see Theorem 6.3.6), the extended limit ! on L1 .0, 1/ can be assumed to be both dilation and C -invariant. Theorem 9.6.9. Let ! be a dilation invariant extended limit on L1 .0, 1/ and let ! denote the Dixmier trace on M1,1 .M, / associated to !. For any dilation invariant extended limit ! on L1 .0, 1/ there exists a dilation and C -invariant extended limit !0 on L1 .0, 1/ such that ! D !0 . Proof. We claim that the mapping t!

1 log.1 C e t /

Z

et

.s, A/ds, 0

t > 0,

Section 9.7 Measurable Operators in L1,1

297

is a Lipschitz function (and, therefore, is uniformly continuous). We have, 0  Z et 1 .s, A/ds log.1 C e t / 0 Z et et 1 e t .e t , A/   D .s, A/ds. log.1 C e t / .1 C e t / log2 .1 C e t / 0 Clearly, 0

e t .e t , A/ 1 t log.1 C e / log.1 C e t /

and

Z

1 log .1 C e t / 2

Hence,

ˇ Z ˇ 1 ˇ ˇ log.1 C e t /

et

0

et

Z

et 0

.s, A/ds

0

.s, A/ds kAkM1,1

1 kAkM1,1 . log.2/

0ˇ   ˇ 1 ˇ .s, A/ds ˇ 1 C kAkM1,1 log.2/

and the claim follows. The assertion now follows from Lemma 9.6.8.

9.7 Measurable Operators in L1,1 The sections of this chapter have characterized the positive Dixmier measurable and positive M -measurable operators in the Dixmier–Macaev operator ideal M1,1 .M, /. We show in this section that Dixmier measurability of arbitrary operators in the ideal L1,1 .L.H /, Tr/ of compact operators can be characterized by eigenvalues. Definition 9.7.1. An arbitrary compact operator A in the Lorentz ideal M1,1 .L.H /, Tr/ is called Tauberian if the limit n X 1 .j , A/ n!1 log.2 C n/

lim

j D0

exists, where .A/ is an eigenvalue sequence of A. It follows from Theorem 9.3.1 that every positive operator A 2 M1,1 .L.H /, Tr/ is Dixmier measurable if and only if it is Tauberian, since n X 1 .j , A/ n!1 log.2 C n/

lim

j D0

298

Chapter 9 Measurability in Lorentz Ideals

exists. Theorem 9.7.5 below shows that an arbitrary operator A 2 L1,1 .L.H /, Tr/ is Dixmier measurable if and only if it is Tauberian. This result cannot be extended to arbitrary compact operators in the Dixmier–Macaev ideal M1,1 .L.H /, Tr/. Example 9.7.6 provides an example of an arbitrary Dixmier measurable operator A 2 M1,1 .L.H /, Tr/ that is not Tauberian. Let X0  L1 .0, 1/ be the linear span of the functions 1 t! log.1 C t /

Z

t

.s, A/ds,

A 2 L1,1 .L.H /, Tr/.

0

Lemma 9.7.2. If ! is an extended limit on L1 .0, 1/, then !.u x/ D !.x/ for every x 2 X0 and for every u > 0. Proof. Since u 1=u is the identity operator, it is sufficient to prove the assertion for u 2 .0, 1/ and for 1 x.t / D log.1 C t /

Z

t

.s, A/ds,

A 2 L1,1 .L.H /, Tr/.

0

It follows from the definition of L1,1 .L.H /, Tr/ that 1 .u x/.t / D log.1 C t =u/ 1 D log.1 C t =u/ D

1 log.1 C t =u/

Z

t =u

.s, A/ds Z Z

0 t 0 t



t.t , A/ .s, A/ds C O log.1 C t /



.s, A/ds C o.1/.

0

The assertion now follows from Remark 6.3.5. Let X be the linear span of u X0, u > 0, and L01 .0, 1/. Lemma 9.7.3. The space X is dilation invariant, that is uX D X, u > 0. For every extended limit ! on L1 .0, 1/, we have !.u x/ D !.x/ for every x 2 X and every u > 0. Proof. The first assertion is trivial. The second assertion follows from Lemma 9.7.2. Lemma 9.7.4. For every extended limit ! on L1 .0, 1/, the functional ! : L1,1 .L.H /, Tr/C ! R (as in Definition 6.3.2) is additive and extends to a Dixmier trace on M1,1 .L.H /, Tr/.

Section 9.7 Measurable Operators in L1,1

299

Proof. It follows from Theorem 6.2.5 that there is a dilation invariant extension !0 of the functional !jX satisfying the condition !.x/ lim sup x.t /, t !1

x 2 L1 .0, 1/.

That is, !0 is a dilation invariant extended limit on L1 .0, 1/. The assertion now follows from Theorem 6.3.3. The following theorem characterizes L1,1 .L.H /, Tr/.

Dixmier measurable operators in

Theorem 9.7.5. Let A 2 L1,1 .L.H /, Tr/. The following conditions are equivalent. (a) A is Dixmier measurable. (b) A is Tauberian, that is the limit n X 1 .k, A/ lim n!1 log.2 C n/ kD0

exists, where .A/ is an eigenvalue sequence of A. Proof. Suppose first that A 2 L1,1 .L.H /, Tr/ is self-adjoint. Set Z t 1 a1 D lim inf ..s, AC /  .s, A //ds, t !1 log.1 C t / 0 Z t 1 a2 D lim sup ..s, AC /  .s, A //ds. t !1 log.1 C t / 0 By Lemma 9.3.6, we can select extended limits !1 , !2 such that   Z t 1 ..s, AC /  .s, A //ds , i D 1, 2. ai D !i .t / 0 Thus, !i .x/ D ai . It follows from Lemma 9.7.4 that !i : L1,1 .L.H /, Tr/C ! R extend to Dixmier traces on M1,1 .L.H /, Tr/. Thus, if A is Dixmier measurable, then a1 D !1 .A/ D !2 .A/ D a2 . Consequently, there exists the limit Z t 1 lim ..s, AC /  .s, A //ds t !1 log.1 C t / 0 or, equivalently, the limit n X 1 ..k, AC /  .k, A // . n!1 log.2 C n/

lim

kD0

300

Chapter 9 Measurability in Lorentz Ideals

By Lemma 5.7.1, n X

..k, AC /  .k, A // D

kD0

n X

.k, A/ C O.1/

kD0

and the assertion follows. Consider now the general case. An operator A is Dixmier measurable if and only if both 0

where k  k2 denotes the Hilbert–Schmidt norm on L2 . Note that every V -modulated operator is automatically Hilbert–Schmidt since kT k2 .1 C kV k1 /kT kmod. Proposition 11.2.2. Let V 2 L.H / be a positive operator. The set of all V -modulated operators is a Banach space. For every A 2 L.H / and for every V -modulated operator T , we have kAT kmod kAk1  kT kmod. Proof. The only statement needing to be proved is the completeness of the linear space of V -modulated operators. Let Tn , n 2 N, be a Cauchy sequence in the space of all V -modulated operators. Note that kTn kmod is a Cauchy sequence in R and that Tn .1 C V /1 is a Cauchy sequence in L2 . Hence, Tn .1 C V /1 ! S 2 L2 . Setting T :D S.1 C V /, we have Tn .1 C tV /1 ! T .1 C tV /1 in L2. Hence, for every t > 0, we have t 1=2 kT .1 C tV /1 k2 D lim t 1=2kTn .1 C tV /1 k2 lim kTnkmod. n!1

n!1

Taking the supremum over t > 0, we infer that T is V -modulated.

340

Chapter 11 Trace Theorems

Our main result on modulated operators is the following theorem on the expectation values of a V -modulated operator with respect to an eigenbasis of V 2 L1,1 . The proof of the theorem is given further below. By the term strictly positive we mean positive with trivial kernel. The commutator subspace of L1,1 , denoted Com.L1,1 /, was introduced in Section 5.7. Theorem 11.2.3. Let 0 < V 2 L1,1 be strictly positive and let T 2 L.H / be a V -modulated operator. If fen g1 nD0 is an eigenbasis for V ordered so that Ven D .n, V /en , n  0, we have (a) T 2 L1,1 and diagfhT en , en ig1 nD0 2 L1,1 . (b) T  diagfhT en , en ig1 nD0 2 Com.L1,1 /. Equivalently, n X

.j , T / 

j D0

n X

hT ej , ej i D O.1/,

n  0,

j D0

where .T / is an eigenvalue sequence of T (see Definition 1.1.10). The theorem has consequences for traces of V -modulated operators. Corollary 11.2.4 forms the centre of our expectation value approach to integration in noncommutative geometry. Corollary 11.2.4. Let 0 < V 2 L1,1 be strictly positive and let T 2 L.H / be a V -modulated operator. If fen g1 nD0 is an eigenbasis for V ordered so that Ven D .n, V /en , n  0, then T 2 L1,1 and (a) for every trace ' : L1,1 ! C,

  '.T / D ' ı diag fhT en , enig1 nD0 .

(b) for every Dixmier trace Tr! : M1,1 ! C,   n X 1 Tr! .T / D ! hT ej , ej i . log.2 C n/ j D0

(c) T is Dixmier measurable if and only if the following limit exists n X 1 hT ej , ej i. lim n!1 log.2 C n/ j D0

(d) the formula

n X

hT ej , ej i D c  log.n/ C O.1/,

n  1,

j D0

holds if and only if '.T / D c for every normalized trace ' : L1,1 ! C.

341

Section 11.2 Modulated Operators

Proof. (a) From Theorem 11.2.3 we have that T diagfhT en , enig1 nD0 2 Com.L1,1 /. Since all traces vanish on the commutator subspace the result follows. (b), (c), (d) From Theorem 11.2.3 we have   n n X X 1 1 1 .j , T / D hT ej , ej i C O . log.2 C n/ log.2 C n/ log.2 C n/ j D0

j D1

Since ! vanishes on c0 then (c) and (d) follow from Theorem 10.1.3, and (b) follows from Theorem 10.1.2 (a). We prove Theorem 11.2.3. Lemma 11.2.5. Let V 2 L.H / be a positive operator. An operator T 2 L2 is V modulated if and only if kTEV Œ0, t 1 k2 D O.t 1=2/,

t > 0,

where EV is the spectral measure of V . Proof. Observe that EV Œ0, t 1  2.1 C tV /1 for t > 0 and, therefore, kTEV Œ0, t 1 k2 2kT .1 C tV /1 k2 2kT kmod  t 1=2 . In order to prove the converse assertion, assume for simplicity that V 1. Let t 2 Œ2k , 2kC1/ for some k  0. kT .1 C tV /1 k2 kTEV Œ0, 2k k2 C

k1 X

kTEV .2j 1 , 2j .1 C tV /1 k2

j D0

O.t 1=2/ C

k1 X

.1 C t 2j 1/1  kTEV .2j 1 , 2j k2

j D0

O.t 1=2/ C O.1/ 

k1 X

.1 C 2kj 1/1  2j=2 D O.t 1=2/.

j D0

Lemma 11.2.6. Let V 2 L.H / be a positive operator and let A 2 L.H /. If T is V -modulated, then TA is jVAj-modulated. Proof. Without loss of generality, kV k1 1 and kAk1 1. Let pn D EV Œ0, 2n  and qn D EjVAj Œ0, 2n  for n  0. We have k.1  pj /Aqk k1 2j kVAqk k1 D 2j k ,

kTpj k2 const  2j=2 .

342

Chapter 11 Trace Theorems

Therefore, kTAqk k2 kTpk Aqk k2 C

k X

kT .pj 1  pj /Aqk k2

j D1

kTpk k2 C

k X

kTpj 1k2  k.1  pj /Aqk k1

j D1

  k X k=2 j=2k D O.2k=2 /. const  2 C 2 j D1

The assertion follows from Lemma 11.2.5. Remark 11.2.7. Let 0 V1 V2 2 L.H /. If T 2 L.H / is V1 -modulated, then T is also V2-modulated. Lemma 11.2.8. Let V 2 L1,1 be a positive operator. Then V is V -modulated. Proof. Since V 2 L2 , it follows, for t > 0, that kVEV Œ0, t 1 k22 D Tr.V 2 EV Œ0, t 1 / D

1 X

Tr.V 2 EV .2n1 t 1 , 2n t 1 //

nD0



1 X

22n t 2 Tr.EV Œ2n1 t 1 , 1//

nD0

D

1 X

22n t 2 nV .2n1 t 1 /.

nD0

Recall that V 2 L1,1 and, therefore, nV .s/ const  s 1 , s > 0. Hence, kVEV Œ0, t 1 k22 const 

1 X

22n t 2  2nC1t D O.t 1/.

nD0

Thus, V is V -modulated by Lemma 11.2.5. Lemma 11.2.9. Let 0 < V 2 L1,1 be strictly positive and let T be V -modulated. For every p > 2, we have T V 1=p 2 Lp=.p1/,1 .

343

Section 11.2 Modulated Operators

Proof. Let S :D T V 1=p and let 2k n < 2kC1. By Lemma 11.2.5, we have

 1 1 X X j 1 j SEV 0, 1 kSEV .2 , 2 k2 2.j C1/=p kTEV .2j 1, 2j k2 n 2 j Dk

j Dk

const 

1 X

2.j C1/=p  2j=2 D const  2k.1=p1=2/ D O.n1=p1=2 /.

j Dk

If feng1 nD0 is an eigenbasis for V ordered so that Ven D .n, V /en , n  0, set Sn x :D

n1 X

hx, ej iSej ,

x 2 H.

j D0

We have 1 X

.j , S/2 D minfkS  Ak22 : rank.A/ ng kS  Sn k22

j Dn

D

1 X j Dn



2 1 D O.n2=p1 /. 0, kSej k2 SE V n 2

Hence, .n, S/ D O.n1=p1 / and the assertion follows. Lemma 11.2.10. Let 0 < V 2 L1,1 be strictly positive and let T V 1=p 2 Lp=.p1/,1 for some p  1. If feng1 nD0 is an eigenbasis for V ordered so that Ven D .n, V /en, n  0, then n X kD0

.k, t . Thus, Z Z d d .1 C t / jf  gj.s/ds D .1 C t / .1 C jsj/2d ds jsj>t jsj>t Z 1 .1 C jsj/2d jsjd 1 d jsj  const. D const  t d t

Thus, the distance from f to the set of all compactly supported functions is positive. Definition 11.3.6. Define the Banach space Lmod;2.Rd / consisting of those functions 1=2 f 2 L2.Rd / with jf j2 2 Lmod;1 .Rd / and norm kf kLmod;2 :D kjf j2 kLmod;1 . Since each function f 2 Lmod;2 .Rd / is square integrable it is interesting to know whether the Fourier transform .Ff / belongs to Lmod;2.Rd /. The answer is negative. Example 11.3.7. For every ˛ 2 .1=4, 0/, we have .Ff˛ /.u/ D const  .1 C juj2 /˛d=2 ,

f˛ .u/ :D juj˛ K˛ .2juj/,

u 2 Rd ,

where K˛ is the Macdonald function (a modified Bessel function of the second kind [1]) for ˛. In particular, f˛ 2 Lmod;2 .Rd / and .Ff˛ / … Lmod;2.Rd /. Proof. We confirm that the functions are related by the Fourier transform. Applying a unitary transform, we obtain, for u 2 Rd , Z Rd

e 2 i hs,ui ds D .1 C jsj2 /˛Cd=2

Z Rd

e 2 i s1juj ds. .1 C jsj2 /˛Cd=2

After substitution sk D vk .1 C s12 /1=2 , 2 k d , we obtain Z Rd

e 2 i hs,ui ds D .1 C jsj2 /˛Cd=2

Z

e 2 i s1juj  .1 C jvj2 /˛d=2 ds1 dv 2 ˛C1=2 d R .1 C s1 / Z Z 1 e 2 i s1juj ds1  .1 C jvj2 /˛d=2 dv D 2 ˛C1=2 1 .1 C s1 / Rd 1

D const  juj˛ K˛ .2juj/. Since Ff˛ 2 L2 .Rd /, it follows that f˛ 2 L2 .Rd /. Now, observe that Z Z j.Ff˛ /.s/j2 ds > const  jsj4˛2d ds D const  .1 C t /4˛d . jsj>t

jsj>t

348

Chapter 11 Trace Theorems

Hence .Ff˛ /2 62 Lmod;1.Rd /. However, as the Macdonald function K˛ decreases exponentially at C1 [1], it follows that f˛2 2 Lmod;1.Rd /. The situation is not improved even if we take a modulated function of compact support. Example 11.3.8. We have f .u/ :D juj1,

.Ff /.u/ D jujd=2 Jd=2.2juj/,

u 2 Rd ,

where Jd=2 is a Bessel function of the first kind [1]. In particular, f 2 Lmod;2 .Rd / is a compactly supported function and .Ff˛ / … Lmod;2 .Rd /. Proof. We confirm that the functions are related by the Fourier transform. For s 2 Rd with jsj 1, we set v D .s2 , : : : , sd / 2 Rd 1 . Observe that jvj .1  s12 /1=2 and, for u 2 Rd , Z Z Z 2 i hs,ui 2 i hs,ui f .s/e ds D e ds D e 2 i s1jujds Rd

Z D

jsj1 Z 1 2 i s1 juj

1

e

Z

D const 

jsj1

jvj.1s12 /1=2 1 1

 dv ds1

.1  s12 /.d 1/=2 e 2 i s1juj ds1.

The latter integral is jujd=2 Jd=2 .2juj/. Evidently, the compactly supported function f 2 is modulated. Since    d C1 , juj ! 1, Jd=2 .2juj/ juj1=2  cos  2juj  4 then .Ff / is strictly of order juj.1Cd /=2 . It follows that .Ff /2 is not modulated.

Laplacian Modulated Operators We associate the Banach algebra Lmod;1 .Rd / to the notion of a Laplacian modulated operator. Lemma 11.3.9. Every Hilbert–Schmidt operator T : L2.Rd / ! L2.Rd / can be represented in the form Z .T x/.u/ D e 2 i hu,sipT .u, s/.Fx/.s/ds, u 2 Rd , (11.7) Rd

for a unique function pT 2 L2.Rd  Rd /.

Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue

349

Proof. The operator T F 1 is Hilbert-Schmidt, and is hence an integral operator with a square integrable kernel, [222, p. 23]. The assertion follows by applying the Plancherel theorem. Definition 11.3.10. We call the function pT in equation (11.7) the symbol of the operator T . The relationship between the kernel and symbol of a Hilbert–Schmidt operator is Z KT .u, v/ D

Rd

e 2 i huv,sipT .u, s/ds,

u, v 2 Rd .

(11.8)

The symbol can be recovered from the operator as follows. Lemma 11.3.11. Let T : L2.Rd / ! L2 .Rd / be a Hilbert–Schmidt operator. If  ,  > 0, is a mollifier, as introduced in Section 10.2, then almost everywhere pT .u, s/ D lim e 2 i hu,siT ..F 1  /e 2 i h,si/.u/, #0

u, s 2 Rd

Proof. We have, for u 2 Rd , that Z 2 i hs,i /.u/ D e 2 i hu,vipT .u, v/ .v  s/dv ! e 2 i hs,uipT .u, s/ T ..F /e Rd

as  # 0. Let  be the Laplacian on Rd . The bounded positive operator .1  /d=2 : L2.Rd / ! L2.Rd / from Example 10.2.16 is used in the following definition. Definition 11.3.12. A bounded operator T : L2.Rd / ! L2.Rd / is said to be Laplacian modulated if it is .1  /d=2-modulated, that is, sup t 1=2 kT .1 C t .1  /d=2/1 k2 < 1, t >0

(11.9)

where k  k2 denotes the Hilbert–Schmidt norm on L2 .L2 .Rd //. Observe that every Laplacian modulated operator is necessarily Hilbert–Schmidt, and as such has a unique symbol. The above definition is not enlightening in practice, however, Laplacian modulated operators are equivalently defined by the following property of their symbols. Lemma 11.3.13. A Hilbert–Schmidt operator T : L2.Rd / ! L2 .Rd / with symbol pT is Laplacian modulated if and only if Z jpT .u, /j2 du 2 Lmod;1 .Rd / Rd

where Lmod;1.Rd / is the Banach algebra of modulated functions.

350

Chapter 11 Trace Theorems

Proof. It follows from Lemma 11.2.5 that T is Laplacian modulated if and only if kTE.1/d=2 Œ0, t 1 k2 D O.t 1=2 /,

t > 0.

(11.10)

The spectral projection of the operator .1  /d=2 is given by the formula Z .E.1/d=2 Œ0, t 1 x/.s/ D .Fx/.u/e 2 i hu,si du, s 2 Rd , .1C4 2juj2 /d=2 t 1

Define a family of projections pt , t > 0, by setting Z .Fx/.u/e 2 i hu,si du, .pt x/.s/ :D juj>t

s 2 Rd .

There exist constants c1 and c2 such that, for every t  1, we have p.c1 t /1=d E.1/d=2 Œ0, t 1  p.c2 t /1=d . It follows from (11.10) that T is Laplacian modulated if and only if kTpt k2 D O.t d=2/. The assertion now follows from the equality Z Z kTpt k22 D jpT .u, s/j2 duds, t > 0. jsj>t

Rd

Remark 11.3.14. The set of symbols of Laplacian modulated operators form a Banach space S mod with norm Z 1=2 Z d=2 2 jpT .u, s/j duds . kpT kmod :D sup.1 C t / t >0

jsj>t

Rd

In fact, Lemma 11.3.13 and its proof show that there is an isometry between the Banach space of Laplacian modulated operators (see Proposition 11.2.2) and the Banach space of their symbols S mod . Definition 11.3.15. An operator T : L2.Rd / ! L2.Rd / is said to be (a) compactly supported if there exist ,

2 Cc1 .Rd / such that M TM D T .

(b) compactly based if there exists  2 Cc1 .Rd / such that M T D T . The proof of the following lemma, as with its equivalent in Section 10.2, is omitted. Lemma 11.3.16. A Laplacian modulated operator T : L2.Rd / ! L2.Rd / is (a) compactly supported if and only if its kernel KT is compactly supported. (b) compactly based if and only if its symbol pT is compactly supported in the first variable.

Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue

351

The Banach space of Laplacian modulated operators is an extension of the set of compactly based pseudo-differential operator of order d . Theorem 11.3.17. If A : Cc1 .Rd / ! Cc1 .Rd / is a compactly based pseudo-differential operator of order d , then (the extensions of) both A and A are Laplacian modulated. Proof. The symbol p of a pseudo-differential operator A of order d belongs to S d . In particular, jp.u, s/j const  hsid . The symbol of a compactly based operator is compactly supported in the first variable. Fix a compact set K  Rd such that p.u, s/ D 0 for u … K. We have Z Z Z jp.u, s/j2 duds const  m.K/ hsi2d ds D O.t d /, jsjt

Rd

jsjt

where m is Lebesgue measure. It follows from Proposition 11.3.13 that A is Laplacian modulated. It follows from Proposition 10.2.19 that A D A1 C A2 with A1 being a compactly supported operator of order d and A2 being a compactly based operator of order 1. Hence, by the above, A1 , a compactly supported operator of order d , is a Laplacian modulated operator. The operator A2 belongs to the Shubin class G 1 . Then A2 2 G 1 . If q is a symbol of A2 , then q.u, s/ const  .1 C juj C jsj/2d , Observe that Z Z 4d .1 C juj C jsj/ du D const  Rd

1

u, s 2 Rd .

.1 C juj C jsj/4d jujd 1 d juj

0

Z

const 

1Cjsj

.1 C jsj/4d jujd 1 d juj 0  Z 1 C juj3d 1 d juj 1Cjsj 3d

D const  .1 C jsj/

.

The mapping s ! .1 C jsj/3d , s 2 Rd , belongs to Lmod;1 .Rd /. It follows from Lemma 11.3.13 that A2 is Laplacian modulated. Hence, A D A1 C A2 is Laplacian modulated.

352

Chapter 11 Trace Theorems

Residues of Laplacian Modulated Operators We define in this section the vector-valued residue of a compactly based Laplacian modulated operator. We show it is an extension of the noncommutative residue of a classical compactly based pseudo-differential operator of order d . Proposition 11.3.18. If T : L2.Rd / ! L2.Rd / is a compactly based Laplacian modulated operator with symbol pT , then Z Z 1 pT .u, s/duds D O.1/, n  0. log.2 C n/ jsjn1=d Rd Proof. It follows from Lemma 11.3.16 that the symbol pT of T is compactly supported in its first variable. Fix a compact set K  Rd such that pT .u, s/ D 0 for u … K. Observe that, for k  0, m.K  fs 2 Rd : e k jsj e kC1g/ D const  m.K/e kd . Using the Cauchy–Schwartz inequality, we have Z Z Z jpT .u, s/jduds D ek jsjekC1

Z

jpT .u, s/jduds Z kd const  m.K/e

Rd

ek jsjekC1

K



ek jsjekC1 1=2 2

Z const  m

Rd 1=2

jpT .u, s/j duds

.K/kpT kmod.

Similarly, Z

Z jsj1

It follows that ˇZ Z ˇ ˇ ˇ jsjt

Rd

Rd

jpT .u, s/jduds const  m1=2.K/kpT kmod.

ˇ Z ˇ pT .u, s/duds ˇˇ

Z jsj1 Rd

jpT .u, s/jduds

Œlog.t / Z

C

Z

X

kD0

ek jsjekC1

Rd

jpT .u, s/jduds

.2 C Œlog.t //  const  m1=2 .K/kpT kmod. The assertion follows by substituting into this inequality the value t D n1=d , n  0, and the fact that log.n1=d / D d 1 log.n/.

Section 11.3 Laplacian Modulated Operators and the Noncommutative Residue

353

Definition 11.3.19. Let T be a Laplacian modulated operator and let pT denote its symbol. The vector-valued linear function Res from the set of all compactly based Laplacian modulated operators into l1 =c0 defined by setting  1

Z Z 1 Res.T / :D , (11.11) pT .u, s/duds log.2 C n/ jsjn1=d Rd nD0 where Œ denotes an equivalence class in l1 =c0 , is called the residue. We show that the map Res extends the noncommutative residue. Lemma 11.3.20. The residue of a compactly based pseudo-differential operator of order d depends only on its principal symbol. Proof. If A1 and A2 are compactly based pseudo-differential operators of order d with the same principal symbol, then A1 A2 is a compactly based pseudo-differential operator with symbol p 2 S d 1 . In particular, jp.u, s/j const  hsid 1 . The symbol of a compactly based operator is compactly supported in the first variable. Fix a compact set K  Rd such that p.u, s/ D 0 for u … K. We have ˇZ Z ˇ Z ˇ ˇ ˇ ˇ p.u, s/duds ˇ const  m.K/ hsid 1 ds < 1. ˇ Rd

Rd

Rd

Hence, Res.A1  A2 / D 0 and, therefore, Res.A1 / D Res.A2 /. We understand the field of scalars to be embedded in `1 =c0 as the equivalence classes of convergent sequences, that is, if ˛ 2 C then ˛ Œfan g1 nD0  where limn!1 an D ˛. Proposition 11.3.21. If A : Cc1 .Rd / ! Cc1 .Rd / is a classical compactly based pseudo-differential operator of order d , then its extension A is Laplacian modulated and Res.A/ D ResW .A/. Here ResW denotes the noncommutative residue of Wodzicki as defined in Definition 10.2.27. Proof. Let p be the principal symbol of the operator A. By Lemma 11.3.20 the value Res.A/ is determined by the equivalence class of the sequence Z Z 1 p.u, s/duds, n  0. log.2 C n/ jsjn1=d Rd

354

Chapter 11 Trace Theorems

Since p is homogeneous of order d , it follows, provided n  1, that   Z Z Z Z s d p.u, s/dsdu D jsj p u, dudt C O.1/. jsj Rd jsjn1=d Rd 1jsjn1=d Observe that s D jsj! with ! 2 Sd 1 and ds D jsjd 1 d jsjd!. Therefore, Z

Z

Z Rd

jsjn1=d

p.u, s/dsdu D

Z

Z Sd 1

Rd

1 D d

Z

Sd 1

jsj1 d jsj C O.1/

1

Z

Rd

n1=d

p.u, !/d!du 

p.u, !/d!du  log.n/ C O.1/. (11.12)

The co-efficient of log.n/ is the noncommutative residue. Hence Res.A/ D ResW .A/. The residue of a pseudo-differential operator is not always a scalar. Proposition 11.3.22. There is a compactly supported pseudo-differential operator Q : Cc1 .Rd / ! Cc1 .Rd / of order d such that 1

 1 1=d  sin.log.log..1 C n/ /// . Res.Q/ D d nD0 Proof. Let q 2 S d be the symbol in Example 10.2.10, that is, q.s/ D jsjd  .sin C cos/.log.log.jsj///,

s 2 Rd , jsj  4.

Let Q0 be the pseudo-differential operator associated to q. Let  2 Cc1 .Rd / be such that kk2 D 1. The operator Q D M Q0 M is compactly supported and Laplacian modulated. Its principal symbol is (see Example 10.2.13) .u, s/ ! j.u/j2 q.s/,

u, s 2 Rd .

Using Lemma 11.3.20, and provided n  4d , Z Z Z Z 2 2 j.u/j q.s/dsdu D j.u/j du  Rd

jsjn1=d

Z

Rd

n1=d

D

q.s/ds C O.1/ 4jsjn1=d

q.jsj/jsjd 1 d jsj C O.1/

4

D log.n/  sin.log.log.n1=d /// C O.1/. We prove one more result, however it will not be used until the section on closed Riemannian manifolds. The next lemma implies that the residue can be defined using any equivalent metric on Rd .

355

Section 11.4 Eigenvalues of Laplacian Modulated Operators

Lemma 11.3.23. Let T : L2.Rd / ! L2.Rd / be a compactly based Laplacian modulated operator with symbol pT and let A : Rd ! Md .Rd / be a positive matrix valued function such that ajsj jA.u/sj bjsj for all u, s 2 Rd for some 0 < a < b. We have Z Z Z Z pT .u, s/dsdu  pT .u, s/dsdu D O.1/, n  0. Rd

jsjn1=d

Rd

jA.u/sjn1=d

(11.13)

Proof. For n  0, the conditions on A ensure that bjsj n implies jA.u/sj n and jA.u/sj n implies ajsj n. Therefore, ˇZ Z ˇ Z Z ˇ ˇ ˇ pT .u, s/dsdu  pT .u, s/dsduˇˇ ˇ d R ajsjn Rd jA.u/sjn Z Z jpT .u, s/jdsdu. Rd

ajsjnbjsj

The rest of the proof follows in an identical manner to that of Proposition 11.3.18.

11.4 Eigenvalues of Laplacian Modulated Operators The following eigenvalue theorem is at the heart of Theorem 11.1.1, and hence it is at the heart of Connes’ Trace Theorem. It associates the residue of a compactly supported Laplacian modulated operator to partial sums of eigenvalues of the operator. Consequently, it provides a new estimate for partial sums of eigenvalues of compactly supported pseudo-differential operators. Using the following estimate, the Lidskii formula for a Dixmier trace (see Theorem 10.1.2 or Theorem 7.3.1) will link Dixmier traces to the residue defined in (11.11). Theorem 11.4.1. Let T : L2.Rd / ! L2.Rd / be a compactly supported Laplacian modulated operator with symbol pT . Then T 2 L1,1 and we have n X j D0

Z .j , T / 

Z Rd

jsjn1=d

pT .u, s/dsdu D O.1/,

n  0,

(11.14)

where .T / is an eigenvalue sequence of T (a sequence of non-zero eigenvalues of T in any order such that their absolute values are decreasing, or the zero sequence if T is quasi-nilpotent, see Definition 1.1.10). The rest of this section proves Theorem 11.4.1. The proof fundamentally relies on the compact support of the Laplacian modulated operator, so that we can use the Dirichlet Laplacian on a bounded domain (which has compact resolvent while the Laplacian on Rd does not) and apply Theorem 11.2.3.

356

Chapter 11 Trace Theorems

Let em , m 2 Zd , be the eigenbasis of .1  /d=2 : L2 .Œ0, 1d / ! L2.Œ0, 1d / where  is the Dirichlet Laplacian. That is, em .u/ :D e 2 i hm,ui, u 2 Œ0, 1d . We extend em as a multi-periodic function on Rd . Lemma 11.4.2. Let T : L2 .Rd / ! L2 .Rd / be a Laplacian modulated operator. For every Schwartz function  2 S.Rd /, we have X kT .em /k2 D O.t d /, t > 0. jmj>t

Proof. Since .F.em //.s/ D .F/.s  m/, s 2 Rd , m 2 Zd , it follows from (11.7) and the Cauchy–Schwartz inequality that Z Z j.T .em //.u/j2 jpT .u, s/j2 j.F/.s  m/jds  j.F/.s  m/jds Rd Rd Z D const  jpT .u, s/j2 j.F/.s  m/jds. Rd

We have used the fact that F is a Schwartz function, so it is integrable. Moreover, j.F/.s/j const  hsi2d . If jv  mj 1=2, then j.F/.s  m/j const  hs  mi2d const  hs  vi2d , and, therefore,

Z

j.T .em //.u/j const  2

Rd

s 2 Rd

jpT .u, s/j2 jhs  vi2d ds.

Define functions f , h 2 Lmod;1 .Rd / by setting Z f .s/ :D jpT .u, s/j2 du, h.s/ :D hsi2d , Rd

(11.15)

s 2 Rd .

For every v with jv  mj 1=2, it follows from (11.15) that kT .em /k2 const  .f ? h/.v/. Therefore, X

kT .em /k2 const 

jmj>t

Z

X Z jmjt

jvmj1=2

.f ? h/dv const 

jvjt 1=2

It follows from Lemma 11.3.2 that f ? h 2 Lmod;1 . Hence, X kT .em/k2 const  kf ? hkLmod;1 O.t d /. jmj>t

.f ? h/dv.

357

Section 11.4 Eigenvalues of Laplacian Modulated Operators

The previous lemma allows us to prove that a compactly supported Laplacian modulated operator is Dirichlet Laplacian modulated. Since the Dirichlet Laplacian has compact resolvent, and by the Weyl law [38], .1  /d=2 2 L1,1 , the following result allows us to use Theorem 11.2.3. Theorem 11.4.3. Let T : L2.Rd / ! L2 .Rd / be a Laplacian modulated operator. If T is compactly supported in the cube Œ0, 1d , then T : L2 .Œ0, 1d / ! L2 .Œ0, 1d / is Dirichlet Laplacian modulated. Proof. Let em , m 2 Zd , be the eigenbasis of the Dirichlet Laplacian indicated above, extended as a multi-periodic function on Rd . Let  : Rd ! R be a positive Schwartz function such that  D 1 on Œ0, 1d . Since T is compactly supported in Œ0, 1d , it follows that T em D T .em/. Therefore, on the Hilbert space L2.Œ0, 1d /, for t > 0 we have

2  1 TE .1/d=2 0, t D 2 D

X

kT em k2

1C4 2jmj2 t 2=d

X

X

kT em k2

jmj.t =4 /1=d

kT .em/k2 D O.t 1/.

jmj.t =4 /1=d

The assertion now follows from Lemma 11.2.5. Before we can use Theorem 11.2.3 though, we need some technical estimates that will allow us to control the difference Z Z X pT .u, s/duds  hT em , em i, t > 0. jsjt

Rd

jmjt

The next two lemmas are purely technical, and the above difference is demonstrated to be bounded in Lemma 11.4.6. Lemma 11.4.4. Let  be a Schwartz function such that  D 1 in Œ0, 1d . We have X e 2 i hu,smi.F/.s  m/ D Œ0,t  .jsj/ C O.ht  jsjid /, t > 0, s 2 Rd , jmjt

uniformly over u 2 Œ0, 1d . Proof. Observe that F is a Schwartz function and, therefore, jFj.s/ D O.hsi2d /, s 2 Rd . For jsj  t , we have ˇ ˇ X ˇ ˇ X 2 i hu,smi ˇ e .F/.s  m/ˇˇ const  hm  si2d const  ht  jsjid . ˇ jmjt

jmjt

358

Chapter 11 Trace Theorems

Since  is Schwartz, for u 2 Œ0, 1d , X

e

2 i hu,smi

Z .F/.s  m/ D

m2Zd

Rd

e

2 i hut ,si

 X

e

2 i ht u,mi

 .t /dt

m2Zd

D .u/. Therefore, X

e 2 i hu,smi .F/.s  m/ D .u/ D 1,

u 2 Œ0, 1d .

m2Zd

Hence, for jsj t and u 2 Œ0, 1d , we have ˇ X ˇ X ˇ ˇ 2 i hu,smi ˇ ˇ const  e .F/.s  m/  1 hm  si2d ˇ ˇ jmjt

jmjt

const  ht  jsjid . Lemma 11.4.5. Let T : L2 .Rd / ! L2 .Rd / be a compactly based Laplacian modulated operator. If pT is the symbol of T then Z Z jpT .u, s/jht  jsjid duds D O.1/, t > 0. Rd

Rd

Proof. For jsj t =2, we have ht  jsjid 2d t d . For jsj > 2t , we have ht  jsjid 2d jsjd . Recall that pT .u, s/ D 0 when u is outside of some compact set K. We have Z Z Z Z d d d jpT .u, s/jht  jsji dsdu 2 t jpT .u, s/jdsdu Rd Rd K jsjt =2 Z Z Z Z jpT .u, s/jdsdu C 2d jsjd jpT .u, s/jdsdu. C K

t =2jsj2t

K

jsj2t

The first term is O.t d log.t // by Lemma 11.3.18. The second term is O.1/ by exactly the same argument as in the proof of Lemma 11.3.18. The third term is bounded by the Cauchy–Schwartz inequality. Lemma 11.4.6. Let T : L2 .Rd / ! L2.Rd / be a Laplacian modulated operator compactly supported in Œ0, 1d . If pT is the symbol of T then Z Z X pT .u, s/duds  hT em , em i D O.1/, t > 0. (11.16) jsjt

Rd

jmjt

Section 11.5 Trace Theorem on Rd

359

Proof. Let  be a Schwartz function such that  D 1 in Œ0, 1d . We have T em D T .em /. Observe that .F.em //.s/ D .F/.s  m/. It follows from (11.7) that Z Z e 2 i hu,smipT .u, s/.F/.s  m/dsdu. hT em , em i D Rd

Rd

Noting that pT .u, s/ D 0 for u … Œ0, 1d , we obtain (using Lemma 11.4.4) ˇZ ˇ Z X ˇ ˇ ˇ pT .u, s/duds  hT em , em iˇˇ ˇ jsjt

Rd

ˇZ ˇ D ˇˇ

jmjt

Z Œ0,1d

Rd

ˇ  X  ˇ 2 i hu,smi pT .u, s/ e .F/.s  m/  Œ0,t .jsj/ dsduˇˇ jmjt

Z

Z

const 

Rd

Œ0,1d

jpT .u, s/j  ht  jsjid dsdu.

The assertion now follows from Lemma 11.4.5. We now have the estimates required to prove Theorem 11.4.1. Proof of Theorem 11.4.1. Without loss we can assume that T is compactly supported in Œ0, 1d . Observe that V :D .1  /d=2 2 L1,1 , where  is the Dirichlet Laplacian, is strictly positive, and from Theorem 11.4.3 the operator T is V -modulated. Theorem 11.2.3 then indicates that T 2 L1,1 .L2 .Œ0, 1d //. Observe that T D P TP where P projects L2.Rd / ! L2.Œ0, 1d /, hence, T 2 L1,1 .L2.Rd // since .T / D .P TP / D .T jL2 .Œ0,1d / /. Let vk , k  0, be a rearrangement of the sequence em , m 2 Zd , according to increasing jmj. For a given n  0, let vn D emn with jmn j n1=d . Theorem 11.2.3 indicates that n X j D0

.j , T / D

n X

hT vj , vj i C O.1/ D

j D0

X

hT em , em i C O.1/.

jmjn1=d

The estimate (11.14) now follows from Lemma 11.4.6.

11.5 Trace Theorem on Rd We prove Theorem 11.1.1, restated as Theorem 11.5.1 below, which associates the noncommutative residue on Laplacian modulated operators with singular traces. Recall from Section 10.1 the definition of the weak-l1 ideal of compact operators, L1,1 :D fA 2 L.H / : .n, A/ D O..1 C n/1 /g,

360

Chapter 11 Trace Theorems

and the Lidskii formula for a Dixmier trace,  Tr! .A/ D !

1  n X 1 , .j , A/ log.2 C n/ nD0

A 2 L1,1 ,

j D0

where ! is an extended limit on l1 (recall from Section 9.7 that, restricted to L1,1 , any extended limit can be used to define a Dixmier trace) and .A/ 2 l1,1 is an eigenvalue sequence of A. The operator A is said to be Dixmier measurable if Tr! .A/ has the same value for every extended limit !. All Dixmier traces are normalized, meaning that   1  1 Tr! diag D 1. n C 1 nD0 We note that an extended limit ! on l1 is a state on the quotient l1 =c0 , !.Œcn / :D !.fcn g1 nD0 /,

fcn g1 nD0 2 l1 .

Theorem 11.5.1 (Trace Theorem). Suppose T : L2.Rd / ! L2.Rd / is a compactly supported Laplacian modulated operator with symbol pT , and 1

 Z Z 1 , pT .u, s/duds Res.T / D log.2 C n/ jsjn1=d Rd nD0 where Πdenotes the equivalence class in l1 =c0 (see Definition 11.3.19). Then T 2 L1,1 and Tr! .T / D !.Res.T //

(a)

for every extended limit ! on l1 . (b) T is Dixmier measurable if and only if Res.T / is scalar valued and then Tr! .T / D Res.T / for every extended limit ! on l1 . '.T / D Res.T /

(c)

for every normalized trace ' on L1,1 if and only if Res.T / is scalar valued and Z Z pT .u, s/dsdu D Res.T /  log.n/ C O.1/, n  1. (11.17) Rd

jsjn1=d

Section 11.5 Trace Theorem on Rd

361

Proof. By Theorem 11.4.1 and Definition 11.3.19, we have that  Res.T / D

1

n X 1 . .j , T / log.2 C n/ nD0

(11.18)

j D0

(a) By Theorem 11.4.1, we have T 2 L1,1 . We apply Theorem 10.1.2 and obtain that  1  n X 1 D !.Res.T //. Tr! .T / D ! .j , T / log.2 C n/ nD0 j D0

(b) It follows from (11.18) and Theorem 10.1.3 that T is measurable if and only Res.T / scalar. (c) Let D :D diag.f.1 C j /1g1 j D0 /. Note that '.T / D Res.T / (with scalar Res.T /) for every normalized trace ' if and only if T  Res.T /D 2 Com.L1,1 /. By Theorem 5.7.6 and Proposition 5.7.5, this occurs if and only if n X

.j , T /  Res.T / 

j D0

n X j D0

1 D O.1/. j C1

(11.19)

By Theorem 11.4.1, this is possible if and only if (11.17) holds. We apply Theorem 11.5.1 to compactly based pseudo-differential operators. Theorem 11.5.2. If A : Cc1 .Rd / ! Cc1 .Rd / is a compactly based pseudo-differential operator of order d , then the extension A 2 L1,1 and Tr! .A/ D !.Res.A// for every extended limit ! on l1 . Proof. It follows from Proposition 11.3.17 that A : L2.Rd / ! L2.Rd / is Laplacian modulated. There exists a compactly supported function  such that A D M A. The operator A0 D AM is compactly supported and the operator A  A0 is compactly based and of order 1. Then Res.A  A0 / D 0 by Lemma 11.3.20. Also A  A0 2 L1 by Corollary 10.2.21. It follows from Theorem 11.5.1 that A0 2 L1,1 and, therefore, A 2 L1,1 . It now follows from Theorem 11.5.2 that Tr! .A/ D Tr! .A0 / D !.Res.A0 // D !.Res.A//. Since Res.A/ is computable from the symbol of A, Theorem 11.5.2 means that we can compute the Dixmier trace of every compactly based pseudo-differential operator,

362

Chapter 11 Trace Theorems

including those that are non-classical. Let us indicate that there are pseudo-differential operators whose Dixmier trace depends on the extended limit !, that is, they are not measurable in the sense of Connes. Corollary 11.5.3. There is a (non-classical) compactly based pseudo-differential operator Q : Cc1 .Rd / ! Cc1 .Rd / of order d such that the value Tr! .Q/ depends on the extended limit !. Proof. From Proposition 11.3.22, there exists a compactly based pseudo-differential operator Q of order d such that 1

 1 1=d  sin.log.log.n /// . Res.Q/ D d nD0 Clearly, Res.Q/ is not a scalar and, thus, Tr! .Q/ depends on the extended limit !. The situation is far different for classical pseudo-differential operators. The corollary below was stated by Connes in [45] for Dixmier traces, however, every normalized trace on L1,1 computes the scalar noncommutative residue. Corollary 11.5.4 (Connes’ Trace Theorem). Suppose A : Cc1 .Rd / ! Cc1 .Rd / is a classical compactly based pseudo-differential operator of order d . Then the extension A 2 L1,1 and, for every normalized trace ' on L1,1 , we have '.A/ D ResW .A/. Here ResW denotes the noncommutative residue of Wodzicki as described in Section 10.2. Proof. By using the same trick as in the proof of Theorem 11.5.2 we may take A, without loss, as compactly supported. From Proposition 11.3.21, Res.A/ D ResW .A/ is a constant and from the proof of that proposition, equation (11.12) in particular, we obtain that (11.17) is satisfied. The assertion now follows from Theorem 11.5.1.

11.6 Trace Theorem on Closed Riemannian Manifolds We now transfer the notion of Laplacian modulated operators and their results to closed Riemannian manifolds. A Hodge–Laplacian modulated operator on a closed manifold, as we define below, is locally a compactly supported Laplacian modulated operator, and the converse is true. We define the residue using the eigenvectors of the Hodge–Laplacian. By showing that Hodge–Laplacian operators are locally Laplacian modulated operators we will use the symbols of Laplacian modulated operators to associate the residue defined using eigenvectors to the integral of symbols over co-disc

Section 11.6 Trace Theorem on Closed Riemannian Manifolds

363

bundles. In this way we can derive Connes’ Trace Theorem and a purely spectral formula for the noncommutative residue of Wodzicki on a closed Riemannian manifold.

Hodge–Laplacian Modulated Operators In this section we introduce Hodge–Laplacian modulated operators and their vectorvalued residue. Suppose X is a d -dimensional closed (and for the purpose of convenient proofs, orientable) Riemannian manifold. From the results in Section 10.3 the Laplace–Beltrami operator associated to a Riemannian metric g admits a unique extension as a negative unbounded operator g : H 2.X/ ! L2.X/ and .1  g /d=2 : L2.X/ ! L2.X/ is a strictly positive compact operator (alternatively, the Laplace–Beltrami operator g has compact resolvent). As a consequence, the Laplace–Beltrami operator on a closed manifold X has crucial spectral properties denied to the Laplacian on Euclidean space. First, there exists an orthonormal basis feng1 nD0 of eigenvectors of the Hodge– Laplacian, g en D 2n en , n  0, ordered such that the eigenvalues 20 21    are increasing. For this section d=2 feng1 . nD0 denotes this ordered eigenbasis of .1  g / Second, by Weyl’s asymptotic formula [38], hn id Cd .1 C n/1 for a constant Cd . Therefore, .1  g /d=2 2 L1,1 belongs to the weak-l1 ideal. These properties are the foundation of spectral geometry [38], and they have markedly strong consequences for the study of the noncommutative residue. Definition 11.6.1. A bounded operator T : L2.X/ ! L2 .X/ is Hodge–Laplacian modulated if it is .1  g /d=2 -modulated, that is, if, for some Riemannian metric g, we have sup t 1=2 kT .1 C t .1  g /d=2 /1 k2 < 1, t >0

where k  k2 denotes the Hilbert–Schmidt norm on L2 .L2 .X//.

364

Chapter 11 Trace Theorems

By the following lemma, Definition 11.6.1 does not depend on the particular choice of Riemannian metric g. Lemma 11.6.2. If T : L2 .X/ ! L2.X/ is as in Definition 11.6.1, then T is .1  g /d=2 -modulated for every Riemannian metric g. Proof. If g1 and g2 are Laplace–Beltrami operators on X with respect to different Riemannian metrics, then .1g1 /d=2 const.1g2 /d=2 . The assertion follows from Remark 11.2.7. A pseudo-differential operator of order d is Hodge–Laplacian modulated. Proposition 11.6.3. If A : C 1 .X/ ! C 1 .X/ is a pseudo-differential operator of order d then (the extension of) A is Hodge–Laplacian modulated. Proof. For brevity, set V :D .1  g /d=2 . The operator B :D AV 1 is a pseudodifferential operator of order 0 and is, therefore, bounded. The operator V belongs to L1,1 and, by Lemma 11.2.8, is Hodge–Laplacian modulated. So then is A D BV by the fact that the set of modulated operators form a left ideal. The Banach space of Hodge–Laplacian modulated operators is a bimodule for the pseudo-differential operators of order 0. Lemma 11.6.4. If T : L2.X/ ! L2 .X/ is a Hodge–Laplacian modulated operator and if A is (the extension of) a pseudo-differential operator of order 0, then both TA and AT are Hodge–Laplacian modulated. Proof. We need only show that TA is Hodge–Laplacian modulated since A has a bounded extension, and the set of Hodge–Laplacian modulated operators is a left ideal of L.L2.X//. For brevity, set V :D .1  g /d=2 . Observe that ŒV , A is a pseudodifferential operator of order d  1 and, therefore, ŒV , A D BV with B a bounded operator. We have, for t > 0, ŒA, .1 C tV /1  D t .1 C tV /1 ŒV , A.1 C tV /1 D .1 C tV /1  B 

tV . 1 C tV

Hence, 1

kT ŒA, .1 C tV /

tV k2 kT .1 C tV / k2  kAk1  1 C tV 1



D O.t 1=2 /

1

and, therefore, kTA.1 C tV /1 k2 kT ŒA, .1 C tV /1 k2 C kT .1 C tV /1 Ak2 D O.t 1=2 /.

Section 11.6 Trace Theorem on Closed Riemannian Manifolds

365

From the spectral properties of the Laplace–Beltrami operator we have that .1  g /d=2 2 L1,1 . Therefore, unlike the similar operator .1  /d=2 on Euclidean space, we can invoke Theorem 11.2.3 directly in order to calculate traces of HodgeLaplacian modulated operators. The difficulty resides in the appropriate notion of a residue of a Hodge-Laplacian modulated operator. We define the residue using the eigenvectors of the Hodge–Laplacian. Lemma 11.6.5. If feng1 nD0 is an ordered eigenbasis of the Hodge–Laplacian and T is a Hodge–Laplacian modulated operator, then n X 1 hT ej , ej i D O.1/, log.2 C n/

n  0.

j D0

Proof. From Theorem 11.2.3 we have that T 2 L1,1 . Hence, .n, T / D O..1 C n/1 /, n  0, and, combining with Theorem 11.2.3 (b), ˇ ˇ n ˇ ˇ n ˇ ˇX ˇ ˇX ˇ ˇ C O.1/ ˇ ˇ hT e , e i .j , T / j j ˇ ˇ ˇ ˇ j D0

j D0



n X

.j , T / C O.1/ D O.log.2 C n//.

j D0

The lemma shows that the following function is well defined. Definition 11.6.6. The vector-valued linear function  Res.T / :D

1

n X 1 , hT ej , ej i log.2 C n/ nD0 j D0

where Œ denotes the equivalence class in `1 =c0 and fen g1 nD0 is an ordered eigenbasis of the Hodge–Laplacian, is called the residue of the Hodge–Laplacian modulated operator T . The next section shows that the residue is a considerable extension of the scalar noncommutative residue on classical pseudo-differential operators of order d .

Symbols of Hodge–Laplacian Modulated Operators Hodge–Laplacian modulated operators are locally Laplacian modulated operators. Proposition 11.6.7. Let X be a closed d -dimensional Riemannian manifold and let .U , h/ be a chart. If T : L2 .X/ ! L2 .X/ is a Hodge–Laplacian modulated operator

366

Chapter 11 Trace Theorems

that is compactly supported in U , then Wh1 T Wh : L2.Rd / ! L2.Rd / is a compactly supported Laplacian modulated where Wh : L2 .U / ! L2.h.U //. Proof. For brevity, set V :D .1  g /d=2 . Let  2 C 1 .X/ be a function compactly supported in U such that M TM D T . By Lemma 11.2.6, T D TM is jVM jmodulated. The latter operator is compactly supported in U . The positive pseudo-differential operator D 2 D Wh1 jVM j2 Wh is compactly supported and p p Wh1 T Wh : L2 .Rd , detgd m/ ! L2.Rd , detgd m/ is D-modulated. Here det g is the determinant of the coordinates of the metric in the chart U , as explained in Section 10.3. Since pthe respective norms of a compactly supported function are equivalent in L2.Rd , detgd m/ and in L2.Rd /, it follows that Wh1 T Wh : L2.Rd / ! L2.Rd / is D-modulated. Since D 2 is a pseudo-differential operator of order 2d , there exists a pseudodifferential operator P of order 0 such that D 2 D .1  g /d=2 P .1  g /d=2 . Since P is order 0, then kP k1 < 1 and D 2 kP k1  .1  /d . It follows that D const  .1  /d=2 by preservation of order by the square root operation on positive operators. The assertion follows by applying Remark 11.2.7. To associate a Hodge–Laplacian modulated operator to a function on the tangent bundle (a symbol) we emulate the usual treatment of pseudo-differential operators. The symbol is defined locally by the restriction to a chart, pulled back to the tangent bundle and then patched together using a partition of unity. We extend Theorem 11.4.1 to a (chart dependent) statement involving the tangent bundle. Lemma 11.6.8. Let T : L2.X/ ! L2.X/ be Hodge–Laplacian modulated operator. If .U , h/ is a chart and if T is compactly supported in U , then n X kD0

Z .k, T / 

Z jsjn1=d

Rd

pW 1 T Wh .x, s/dxds D O.1/, h

n  0,

where .T / is an eigenvalue sequence of T . Proof. By Proposition 11.6.7, Wh1 T Wh is Laplacian modulated and it is clear that it is compactly supported. The assertion now follows from Theorem 11.4.1 since eigenvalues are invariant under the isometry Wh : L2.h.U // ! L2.U /.

367

Section 11.6 Trace Theorem on Closed Riemannian Manifolds

Let .Ui , hi /, 1 i N , be an atlas of the manifold X. Take a smooth partition of unity j , 1 j M , such that for every pair .j , k/ with j k ¤ 0 both j and k are compactly supported in some Ui . We will always use such a partition of unity and we label it by ‰ D f j gM j D1 . With respect to such a partition of unity we are able to define a coordinate dependent symbol. Definition 11.6.9. If T : L2 .X/ ! L2.X/ is a Hilbert–Schmidt operator, then define .‰,g/ 2 L2.X  Rd / by the formula a coordinate dependent symbol pT .‰,g/ .u, s/ pT

:D

M X

q

Ui .u/ det.g 1 /.hi .u//pW 1M hi

j ,kD1

where i is chosen so that the support of

j

and

k

j

TM

k

Whi .hi .u/, s/,

belong to Ui .

Definition 11.6.10. An operator T : L2.X/ ! L2 .X/ is called localized if M TM 2 L1 for every disjointly supported , 2 C 1 .X/. For a given partition ‰ of unity, an operator is called ‰-localized if M L1 for every pair .j , k/ with j k D 0.

j

TM

k

2

Theorem 11.6.11. Let .X, g/ be a closed d -dimensional Riemannian manifold. Suppose that the Hodge–Laplacian modulated operator T : L2 .X/ ! L2 .X/ is ‰localized. We have Z Z n X .m, T /  pT.‰,g/.u, s/duds D O.1/, n  0, jsjn1=d

mD0

X

where .T / is an eigenvalue sequence of T . Proof. Every operator M j TM k is Hodge–Laplacian modulated by Lemma 11.6.4. If j k ¤ 0, then M j TM k is compactly supported in some Ui . By Proposition 5.7.5, we have n X mD0

.m, T / D

n M X X

.m, M

j

TM

k

/ C O.1/.

j ,kD1 mD0

The assertion now follows from Lemma 11.6.8. We define an equivalence class of symbols and show that, for localized Hodge– Laplacian modulated operators, the residue is defined by the equivalence class. The next result says that every localized Hodge–Laplacian modulated operator can be assigned a coordinate and metric independent “principal” symbol.

368

Chapter 11 Trace Theorems

Corollary 11.6.12. Let X be a d -dimensional closed Riemannian manifold and let g1 , g2 be Riemannian metrics on X. If T is a localized Hodge–Laplacian modulated operator, then Z Z Z Z .‰1 ,g1 / .‰ ,g / pT .u, s/duds  pT 2 2 .u, s/duds D O.1/ jsjn1=d

and Z

Z

jsjn1=d

jsjn1=d

X

.‰1 ,g1 /

X

pT

X

Z Z .u, s/duds  X

.‰1 ,g1 /

1=2 jg1 .u/sjn1=d

pT

.u, s/duds D O.1/

for partitions ‰1 , ‰2 of unity. Proof. The first assertion follows from Theorem 11.6.11 since eigenvalues are metric and coordinate independent. The second assertion follows from Lemma 11.3.23. The differences in the lemma define equivalence relations on coordinate dependent symbols. That is, two coordinate dependent symbols are equivalent if they satisfy the displays in Corollary 11.6.12. We will only be concerned with coordinate dependent symbols up to this equivalence. Definition 11.6.13. If T : L2 .X/ ! L2 .X/ is a localized Hodge–Laplacian modu.‰,g/  of any coordinate dependent symbol lated operator, then the equivalence class ŒpT .‰,g/ is called the symbol of T (that all coordinate dependent symbols generate the pT same class is shown in Corollary 11.6.12 above). We may fix a metric g and we may take any coordinate dependent symbol to act as a representative for the symbol when discussing localized Hodge–Laplacian modulated .‰,g/ operators. Hence, we let pT denote a coordinate dependent symbol pT , dropping explicit reference to the coordinates and the metric. For a localized Hodge–Laplacian modulated operator we characterize its residue (as in Definition 11.6.6, which was a purely spectral definition) in terms of its symbol. Theorem 11.6.14. Let T : L2.X/ ! L2.X/ be a localized Hodge–Laplacian modulated operator. Then the residue of T depends only on its symbol pT , and  Res.T / D

1 log.2 C n/

Z

1

Z jsjn1=d

X

pT .u, s/duds

where Πdenotes the equivalence class in l1 =c0 .

(11.20) nD0

369

Section 11.6 Trace Theorem on Closed Riemannian Manifolds

Proof. If fej g1 j D0 is an ordered eigenbasis for the Hodge–Laplacian, then Theorem 11.2.3 proves that n X

hT ej , ej i D

j D0

n X

.j , T / C O.1/.

j D0

If pT is any coordinate dependent symbol in the equivalence class associated to T , then Theorem 11.6.11 and Corollary 11.6.12 prove that n X

Z .j , T / D

j D0

Z jsjn1=d

X

pT .u, s/duds C O.1/.

The assertion follows from Definition 11.6.6. It also follows that, if pT is well defined as a function on the tangent bundle T  X, then equivalently the residue is defined in terms of co-disc bundles " 1 # Z 1 Res.T / D pT .v/dv (11.21) log.2 C n/ D .n1=d /X nD0 where dv is the volume form on T  X. Pseudo-differential operators on manifolds are localized operators. We can now show that the residue of a classical pseudo-differential operator of order d is Wodzicki’s noncommutative residue. Corollary 11.6.15. If A : C 1 .X/ ! C 1 .X/ is a classical pseudo-differential operator of order d , then the extension A is a localized Hodge–Laplacian modulated operator with Res.A/ D ResW .A/ where ResW is Wodzicki’s noncommutative residue (as in Section 10.3). Proof. That A is localized follows from the definition of a pseudo-differential operator (see Definition 10.3.2). Suppose A is also of order m < d . Then A is trace class and, by Definition 11.6.6, Res.A/ D 0. Thus, the residue depends only on the principal symbol pd of A. It follows from Theorem 11.6.14 that  Res.A/ D

1 log.2 C n/

Z 1jsjn1=d

1

Z X

pd .u, s/duds

. nD0

370

Chapter 11 Trace Theorems

Setting s D jsj! with ! 2 Sd 1 , we have pd .u, s/ D jsjd pd .u, !/ if jsj  1. Therefore, Z Z pd .u, s/duds X

1jsjn1=d

Z Z

D

Z ZS

X

D

1 d

Z

X

d 1

Sd 1

n1=d

d

jsj

d 1

pd .u, !/jsj

 d jsjd! du

1

pd .u, !/d!du  log.n/ C O.1/.

(11.22)

Hence, Res.A/ D

1 d

Z Z X

Sd 1

pd .u, !/d!du D

1 d

Z SX

pd .s/ds,

where ds is the volume form on S  X. The last equality in the display follows by previous remarks as the principal symbol of A is well defined on the tangent bundle, [221]. Let us state what this means as an explicit spectral formula for the noncommutative residue. The next assertion follows immediately from Corollary 11.6.15 and the definition of the residue on Hodge–Laplacian modulated operators, Definition 11.6.6. Corollary 11.6.16 (Spectral formula for the noncommutative residue). Let A : C 1 .X/ ! C 1 .X/ be a classical pseudo-differential operator of order d , and let feng1 nD0 be an ordered eigenbasis of the Hodge–Laplacian g . Then n X 1 ResW .A/ D lim hAej , ej i n!1 log.2 C n/ j D0

where ResW is Wodzicki’s noncommutative residue.

Traces of Hodge–Laplacian Modulated Operators We use Corollary 11.2.4 to identify traces of Hodge–Laplacian modulated operators with their residue. Theorem 11.6.17 (Trace Theorem on manifolds). Let T : L2.X/ ! L2.X/ be a Hodge–Laplacian modulated operator. Then T 2 L1,1 and (a) Tr! .T / D !.Res.T // for every extended limit ! on l1 .

371

Section 11.6 Trace Theorem on Closed Riemannian Manifolds

(b) T is Dixmier measurable if and only if Res.T / is scalar valued and then Tr! .T / D Res.T / for every extended limit ! on l1 . (c) '.T / D Res.T / for every normalized trace ' on L1,1 if and only if Res.T / is scalar valued and satisfies the condition n X

hT ej , ej i D Res.T /  log.n/ C O.1/,

n  1,

j D0

(or

Z

Z jsjn1=d

X

pT .u, s/duds D Res.T /  log.n/ C O.1/,

n  1,

(11.23)

if T is localized and pT is the symbol of T .) Proof. All statements follow immediately from the definition of the residue, Corollary 11.2.4, and Theorem 11.6.14. The first corollary of Theorem 11.6.17 is that the Dixmier trace of any pseudodifferential operator of order d can be computed from local symbols. Corollary 11.6.18. Let A : C 1 .X/ ! C 1 .X/ be a pseudo-differential operator of order d . Then the extension A 2 L1,1 is Hodge–Laplacian modulated and localized and Tr! .A/ D !.Res.A// for every extended limit ! on l1 . The second corollary is that there are non-measurable pseudo-differential operators of order d . Hence, the set of pseudo-differential operators of order d does not have a unique trace. Corollary 11.6.19. There exists a (non-classical) pseudo-differential operator Q : C 1 .X/ ! C 1 .X/ of order d such that the value Tr! .Q/ depends on the extended limit !. Proof. Select a chart .U , h/ and let the operator Q be compactly supported in U . In local coordinates, set Q to be the operator constructed in Corollary 11.5.3.

372

Chapter 11 Trace Theorems

The third corollary of Theorem 11.6.17 is Connes’ Trace Theorem for manifolds. The previous corollary shows that the qualifier classical cannot be omitted from the following statement. Corollary 11.6.20 (Connes’ Trace Theorem). Let A : C 1 .X/ ! C 1 .X/ be a classical pseudo-differential operator of order d . We have A 2 L1,1 and '.A/ D ResW .A/ for every normalized trace ' on L1,1 where ResW is Wodzicki’s noncommutative residue. Proof. The assertion follows from Theorem 11.6.17 (c) as, from the proof of Corollary 11.6.15 (in particular, the equation (11.22)), we have that Z Z p.u, s/duds D ResW .A/  log.n/ C O.1/, n  1, jsjn1=d

X

where p is the symbol (or equivalently, principal symbol) of the operator A.

11.7 Integration of Functions Laplacian modulated operators are a wider class than pseudo-differential operators. We show in this section that the operator Mf .1  /d=2 : L2.Rd / ! L2.Rd /, explicitly x.t / ! f .t /..1  /d=2 x/.t /,

x 2 L2.Rd /,

(11.24)

is Laplacian modulated for any square integrable function f (here .1  /d=2 is the bounded positive operator described in Example 10.2.16). Theorem 11.5.1 is used to compute the singular trace of the operator (11.24) when f is compactly supported. It turns out that this computes the Lebesgue integral of the function f . For a locally bounded measurable function f on Rd define an (unbounded) normal operator Mf by setting .Mf x/.t / :D f .t /x.t /,

x 2 L1 .Rd /, x has compact support .

Lemma 11.7.1. The operator Mf .1  /d=2 is Hilbert–Schmidt if and only if f 2 L2.Rd /. Proof. First, let f 2 L2.Rd /. Set g.t / :D .1 C 4 2jt j2 /d=2 , t 2 Rd . We have .1  /d=2 F 1 D F 1 Mg . Thus, Z d=2 1 .Mf .1  / F x/.u/ D f .u/g.s/e 2 i hu,si x.s/ds. (11.25) Rd

373

Section 11.7 Integration of Functions

Hence, Mf .1  /d=2F 1 is an integral operator with square integrable kernel (that is, a Hilbert–Schmidt operator). Since the Fourier transform is a unitary operation, then the operator Mf .1  /d=2 is Hilbert-Schmidt. Conversely, let Mf .1  /d=2 be a Hilbert–Schmidt operator. Without loss of generality, f  0. Fix a cube K and let fn :D minfn, f K g 2 L2 .Rd /, n  0. It follows from (11.25) that const  kfn k2 D kMfn .1  /d=2 k2 kMf .1  /d=2 k2 . Taking the limit n ! 1, we obtain that kf K k2 const  kMf .1  /d=2 k2. Since the cube K is arbitrarily large, the assertion follows.

Modulation is Not Preserved by Taking Adjoints The Laplacian modulated operators form a left ideal inside the Hilbert–Schmidt operators. They do not form a two-sided ideal; we use operators of the form (11.24) to give an explicit example where the adjoint of a Laplacian modulated operator is not Laplacian modulated. Lemma 11.7.2. If g.s/ :D .1 C 4 2 jsj2 /d=2, s 2 Rd , then pMf .1/d=2 .u, s/ D f .u/g.s/, Z e 2 i hu,vig.s  v/.F 1 f /.v/dv. p.1/d=2 Mf .u, s/ D Rd

Proof. We have .1  /d=2 D F 1 Mg F and, therefore, Z f .u/g.s/e 2 i hu,si .Fx/.s/ds, .Mf .1  /d=2 x/.u/ D Rd

x 2 L2.Rd /,

and the first assertion follows from the definition of the symbol. Similarly, for x 2 L2.Rd /, Z d=2 ..1  / Mf x/.u/ D e 2 i hu,vig.v/.F.f x//.v/dv Rd Z Z D e 2 i hu,vig.v/.Fx/.s/.Ff /.v  s/dvds d d R R Z  Z 2 i hu,si D e .Fx/.s/ e 2 i hu,vsi g.v/.Ff /.v  s/dv ds. Rd

Rd

374

Chapter 11 Trace Theorems

Hence, Z p.1/d=2Mf .u, s/ D Z D Z D

Rd

Rd

Rd

e 2 i hu,vsig.v/.Ff /.v  s/dv e 2 i hu,vig.v C s/.Ff /.v/dv e 2 i hu,vig.s  v/.F 1 f /.v/dv.

The next result provides an example where right multiplication by a square integrable function does not yield a Laplacian modulated operator but left multiplication does. It also provides a condition on the Fourier transform of a square integrable function so that both left and right multiplication yield a Laplacian modulated operator. The Banach space Lmod;2 .Rd /, a subset of L2.Rd /, was introduced in Section 11.3, see Definition 11.3.6. Proposition 11.7.3. We have that (a) Mf .1  /d=2 is Laplacian modulated for every f 2 L2.Rd /. (b) there exists f 2 Lmod;2 .Rd / such that the operator .1/d=2 Mf is not Laplacian modulated. (c) .1/d=2Mf is Laplacian modulated provided that Ff 2 Lmod;2.Rd /, where F denotes the Fourier transform. Proof. Define a function g by setting g.t / :D .1 C 4 2jt j2 /d=2 ,

t 2 Rd .

Recall that jgj2 2 Lmod;1.Rd /. (a) Let p be the symbol of Mf .1  /d=2 . It follows from Lemma 11.7.2 that Z Rd

jp.u, /j2 du D kf k22jgj2 2 Lmod;1 .Rd /.

By Lemma 11.3.13, the operator Mf .1  /d=2 is Laplacian modulated. (b) Let f˛ be the function constructed in Example 11.3.7. We have F 1 f˛ D g 1C2˛=d and jf˛ j2 2 Lmod;1.Rd /. If p is a symbol of the operator .1  /d=2 Mf , then

375

Section 11.7 Integration of Functions

it follows from Lemma 11.7.2 that Z jp.u, /j2 du D jgj2 ? jF 1 f˛ j2 Rd

D g 2 ? g 2C4˛=d  const  g 2C4˛=d … Lmod;1 .Rd /. By Lemma 11.3.13, the operator .1  /d=2 Mf is not Laplacian modulated. (c) If p is the symbol of the operator .1  /d=2 Mf , then it follows from Lemma 11.7.2 and Lemma 11.3.2 that Z jp.u, /j2 du D jgj2 ? jF 1 f j2 2 Lmod;1.Rd /. Rd

Hence, the operator is Laplacian modulated by Lemma 11.3.13.

Integration of Compactly Supported Square Integrable Functions on Rd If f 2 C 1 .Rd / is a compactly supported function, then Mf .1  /d=2 is a classical pseudo-differential operator of order d . It has been noted by many authors that Connes’ original trace theorem implied Z Vol .Sd 1 /  f .u/du, f 2 Cc1 .Rd /. Tr! .Mf .1  /d=2 / D d.2/d Rd R This led a Dixmier trace Tr! to be denoted by a symbol (under the assumption of measurability in the sense of Connes) and termed integration. In fact, from Corollary 11.5.4, Z Vol .Sd 1 / '.Mf .1  /d=2 / D  f .u/du, f 2 Cc1 .Rd /, d.2/d Rd for every normalized trace ' on L1,1 . We improve on the condition that f be smooth. Theorem 11.7.4. If f 2 L2.Rd / is compactly supported, then Mf .1/.d C1/=2 2 L1 . Proof. The proof of this well-known result [222, Theorem 4.5], is provided, amongst similar results, in Chapter 4 of [222]. Theorem 11.7.5. If f 2 L2.Rd / is a compactly supported function, then Mf .1  /d=2 2 L1,1 and '.Mf .1  /d=2 / D for every normalized trace ' on L1,1 .

Vol .Sd 1 /  d.2/d

Z Rd

f .u/du

376

Chapter 11 Trace Theorems

Proof. Let  2 Cc1 .Rd / be such that f  D f . By Proposition 11.7.3, the operator Mf .1  /d=2 is Laplacian modulated. It follows from Lemma 11.6.4 that Mf .1  /d=2M is a compactly supported Laplacian modulated operator. By Theorem 11.5.1 we have that Mf .1  /d=2M 2 L1,1 . Define the pseudodifferential operator Q :D ŒM , .1  /d=2 , which, by Example 10.2.13 is of order d  1. Since Q is order d  1 then there exists a zero order pseudo-differential operator R such that Q D .1  /.d C1/=2 R. We have Mf .1  /d=2  Mf .1  /d=2 M D Mf Q D Mf .1  /.d C1/=2R. By Theorem 11.7.4 the operator Mf .1  /.d C1/=2 is trace class and by Proposition 10.2.15 the operator R has a bounded extension. Hence, Mf .1  /.d C1/=2 R is trace class and Mf .1  /d=2  Mf .1  /d=2 M 2 L1 . Hence, Mf .1  /d=2 2 L1,1 and, for any trace ' on L1,1 , '.Mf .1  /d=2 / D '.Mf .1  /d=2 M /,

(11.26)

since ' vanishes on L1 . Denote by p the symbol of the operator Mf .1/d=2 and by r the symbol of the operator Mf .1  /d=2 M . Then .p  r /.u, s/ D f .u/q.u, s/, where q is the symbol of the operator Q. Hence, ˇZ Z ˇ Z Z ˇ ˇ ˇ ˇ const  .p  r /.u, s/dsdu jf .u/j.1 C jsj/d 1 dsdu D O.1/. ˇ ˇ Rd

jsjt

Rd

Rd

This shows that the symbol r satisfies (11.17), since p is easily checked to satisfy (11.17). It also shows that, by the definition of the vector-valued residue of compactly based Laplacian modulated operators, Res.Mf .1  /d=2 / D Res.Mf .1  /d=2 M /.

(11.27)

Since r satisfies (11.17) and Mf .1  /d=2M is compactly supported then, by Theorem 11.5.1, '.Mf .1  /d=2M / D Res.Mf .1  /d=2 M /.

(11.28)

The assertion of the theorem now follows from (11.26), (11.27) and (11.28).

Integration of Square Integrable Functions on a Closed Manifold Suppose that X is a d -dimensional closed Riemannian manifold with metric g. If f 2 L2 .X/, then define Mf : L1 .X/ ! L2.X/ by Mf x.u/ :D f .u/x.u/,

377

Section 11.7 Integration of Functions

x 2 L1 .X/. In this section g denotes the Laplace–Beltrami operator (see Definition 10.3.3) and .1  g /d=2 2 L1,1 is the compact operator described in Example 10.3.5 and Section 11.6. We start with the following analog of Theorem 11.7.4. Theorem 11.7.6. If f 2 L2 .X/, then Mf .1  g /.d C1/=2 2 L1.L2 .X//. Proof. We refer the reader to Chapter 4 of [222]. Lemma 11.7.7. If f 2 L2.X/, then the operator Mf .1  g /d=2 is localized. Proof. If ,

2 C 1 .X/ are such that 

D 0, then

M .Mf .1  g /d=2/M D Mf  Œ.1  g /d=2 , M . By Example 10.2.13, the operator Œ.1g /d=2 , M  is a pseudo-differential operator of order d  1. It follows that M .Mf .1  g /d=2 /M D Mf  .1  g /.d C1/=2  P where P is a pseudo-differential operator of order 0. Since P is bounded, the assertion follows from Theorem 11.7.6. Proposition 11.7.8. If f 2 L2.X/, then the operator Mf .1  g /d=2 is Hodge– Laplacian modulated. Proof. Fix a chart .U , h/. Without loss of generality, f is compactly supported in U . Select  2 C 1 .X/ compactly supported in U and such that f  D f . We have Wh1 .Mf .1  g /d=2 M /Wh D Mf ıh1 .1  /d=2  P , where Mf ıh1 .1  /d=2 is a Laplacian modulated operator (by Proposition 11.7.3) and  P D .1  /d=2  Wh1 .1  g /d=2 Wh  Mıh1 is a pseudo-differential operator of order 0. By Lemma 11.6.4, Wh1 .Mf .1  g /d=2 M /Wh is a compactly supported Laplacian modulated operator. By Proposition 11.6.7, Mf .1  g /d=2 M is Hodge–Laplacian modulated.

378

Chapter 11 Trace Theorems

Recall that Œ.1  g /d=2 , M1  is a pseudo-differential operator of order d  1 (see Example 10.2.13 for the local result). Define a pseudo-differential operator P of order 1 by setting Œ.1  g /d=2 , M1  D .1  g /d=2 P . An easy inductive argument shows that .1  g /d=2 D .1  g /d=2 P d C1 C

d X

M1 .1  g /d=2 P k C .1  g /d=2 M P k .

kD0

Since Mf M1 D 0, it follows that Mf .1  g /d=2 D

d X

.Mf .1  g /d=2 M /P k C Mf .1  g /d=2P d C1 .

kD0

By the previous paragraph, Mf .1  g /d=2 M is Hodge–Laplacian modulated. By Lemma 11.6.4, so is .Mf .1  g /d=2 M /P k for k  0. Note that Mf .1  g /d=2P d C1 D .Mf .1  g /.d C1/=2/  .1  g /d=2  Q with Q being a pseudo-differential operator of order 0. The operator Mf .1  g /d=2 P d C1 is hence Hodge–Laplacian modulated by the left ideal property of the set of Hodge–Laplacian modulated operators and by Lemma 11.6.4. Thus, Mf .1  g /d=2 is Hodge–Laplacian modulated. Proposition 11.7.9. If f 2 L2.X/, then d=2

Res.Mf .1  g /

Vol.Sd 1 / /D  d.2/d

Z f .u/du. X

Proof. Fix some chart .U , h/. Without loss of generality, f is compactly supported in U . Lemma 11.7.7 and Lemma 11.7.8 provide that Mf .1  g /d=2 is a localized Hodge–Laplacian modulated operator. Its symbol p (the equivalence class in the sense of Definition 11.6.13) is generated by the function on U  Rd , q p.u, s/ D det.g 1 /.h.u//f .u/.2/d jg 1=2 .h.u//sjd ,

379

Section 11.7 Integration of Functions

see Example 10.3.5, and Z

Z jsjn1=d

p.u, s/duds X

1 D .2/d

Z Rd

Z jsjn1=d

f ı h1 .t /jg 1=2 .t /sjd dsdt C O.1/.

Using the equivalences in Corollary 11.6.12, Z Z 1 D f ı h1 .t /jg 1=2 .t /sjd dsdt C O.1/. .2/d Rd jg 1=2.t /sjn1=d Using the change of coordinates in s, Z Z p 1 1 d f ı h .t /jsj detg.t /dsdt C O.1/ D .2/d Rd jsjn1=d Z  Z  p 1 1 d f ı h .t / detg.t /dt  jsj ds C O.1/ D .2/d Rd jsjn1=d Z Vol.Sd 1 / f .u/du  log.n/ C O.1/. (11.29) D d.2/d X

Integration of a square integrable function on a closed Riemannian manifold is recovered by any trace. Theorem 11.7.10. If f 2 L2 .X/, then Mf .1  g /d=2 2 L1,1 and '.Mf .1  g /d=2 / D

Vol.Sd 1 /  d.2/d

Z f .u/du X

for every normalized trace ' : L1,1 ! C. Proof. The first assertion follows from Proposition 11.7.8 and Lemma 11.2.9. The second assertion follows from Proposition 11.7.9, equation (11.29), and Theorem 11.6.17 (c). The condition that f is square integrable cannot be dropped. Adapting the proof of Lemma 11.7.1 it can be shown that Mf .1  g /d=2 is Hilbert–Schmidt if and only if f 2 L2.X/. It follows that Mf .1  g /d=2 2 L1,1 if and only if f 2 L2.X/, see also [151].

380

Chapter 11 Trace Theorems

11.8 Notes Origin of the Approach Connes’ original trace theorem is usually where the foundation of the noncommutative residue ends and applications begin (the rationale derived from the theorem is that Dixmier traces are the “noncommutative residue” for noncommutative versions of pseudo-differential operators). We thought, however, there was more to the story behind Connes’ Trace Theorem. The authors, with Nigel Kalton and Denis Potapov, questioned whether Dixmier traces were the only singular traces that identify with the noncommutative residue. Results of Wodzicki indicated that they were not. We also questioned whether the Dixmier trace of an arbitrary (a not necessarily classical) pseudo-differential operator of order d had a similar formula. The answers we found were reported in [126], which is the origin of the approach to trace theorems covered in this chapter. Laplacian Modulated Operators A function L2 .Rd  Rd / considered as the symbol of a Hilbert–Schmidt operator is a common view of extending the notion of a pseudo-differential operator [58]. The Banach space of symbols associated to the Banach space of Laplacian modulated operators and the concept of a Laplacian modulated operator were introduced by the authors in [126]. The extension of the noncommutative residue from order d pseudo-differential operators to compactly-based Laplacian modulated operators was developed in [126]. There are various examples of extending the noncommutative residue for specific pseudo-differential operator classes, e.g. [17]. The advantage of the estimate in the eigenvalue theorem, Theorem 11.4.1, is the ability to calculate traces, as shown by the trace theorem, Theorem 11.5.1. Classical texts on pseudodifferential operators do not usually consider the ideal L1,1 explicitly. It features naturally in the classical approach to spectral asymptotics of strictly positive hypoelliptic operators, see [221, §30 ], but not for non-elliptic operators. Estimates of the singular values of operators of the form Mf g./, are known as Cwikel estimates (see [13, 57], [222, Chapter 4], [251]). The estimates for singular values that we know of which are the closest to the modulation condition have appeared in analysis using Gabor frames, see [110]. The trace theorem, Theorem 11.5.1, is an extension of Connes’ Trace Theorem. Connes’ statement involving Dixmier traces first appeared in [45, Theorem 1]. Dixmier introduced his traces much earlier in [62]. The commutator subspace has been used previously to study spectral forms of the Dixmier traces, e.g. [6,89]. Outside of [6] the proof we have given of Connes’ original theorem is distinct from others, as it does not rely on the properties of the noncommutative residue (that it is the unique positive trace on classical pseudo-differential operators of order d ) or on -function techniques. For the standard method of proving Connes’ original theorem, following [45], see the monograph [100, Theorem 7.18] or [4, §4.6]. We developed the result for non-measurable pseudo-differential operators in [126]. Spectral Formulas for the Noncommutative Residue The idea of the noncommutative residue as a spectral object appeared in [89, p. 359] and [6, Corollary 2.14], where the noncommutative residue was equated with a formula involving only the eigenvalues of a classical pseudo-differential operator of order d . The formula involving expectation values (Corollary 11.6.16) was obtained as a corollary in [126] of Nigel

Section 11.8 Notes

381

Kalton’s results on modulated operators. The extension of the noncommutative residue to a vector-valued residue on any Hodge–Laplacian modulated operator by using the expectation values (Definition 11.6.6), and the subsequent ability to transfer this residue to (unital) noncommutative geometry, as discussed in the next chapter, is a new result. Integration of Functions For the observation that Connes’ Trace Theorem can be used to recover the integral of a smooth function see, e.g. [100, Corollary 7.21], [9, §1.1], [143, p. 98], [189]. Connes used the function residue to derive the integral on L1 .X / in [47]. Theorem 11.7.10 is a stronger variant of [151, Theorem 2.5]. In the cited paper we proved the statement of Theorem 11.7.10 for Dixmier traces associated to -function residues. The proof employing the methods of Hodge–Laplacian modulated operators is different, and yields the result that integration of square integrable functions on closed manifolds is independent of the trace used. To investigate how traces can recover integration of functions that are not square integrable, the product Mf .1  g /d=2 needs to be replaced by alternative symmetrized versions such as .1  g /d=4 Mf .1  g /d=4 . We refer the interested reader to [151].

Chapter 12

Residues and Integrals in Noncommutative Geometry 12.1 Introduction In this chapter we define the noncommutative residue for an arbitrary unital spectral triple in the noncommutative, or quantum, calculus of Alain Connes, and we show how the noncommutative residue in noncommutative geometry is analogous to its classical counterpart. The quantum calculus is a direct analog of the pseudo-differential operator calculus of a closed Riemannian manifold. The spectral formula for the scalar noncommutative residue on a manifold obtained in Corollary 11.6.16 in Chapter 11 suggests the following construction. If H is a separable Hilbert space and D : Dom.D/ ! H is an unbounded selfadjoint operator with compact resolvent (such a pair .H , D/ is commonly called an unbounded Fredholm module), define  1

n X 1 ResD .A/ :D , A 2 M1,1 hAej , ej i log.2 C n/ nD0 j D0

where Œ denotes an equivalence class in l1 =c0 and fej g1 j D0 is an orthonormal basis of H such that Dej D j ej for the eigenvalues j0j j1 j j2 j    of D. We refer to the basis for H as in the last sentence, which is guaranteed by the fact that D has compact resolvent, as an ordered eigenbasis for the operator D. The first task of this chapter is to show that the noncommutative residue in noncommutative geometry is analogous to its Riemannian geometry counterpart. An unbounded Fredholm module has dimension d > 0 if hDid :D .1 C D 2/d=2 belongs to the weak-l1 ideal of compact operators L1,1 :D fA 2 L.H / : .A/ 2 l1,1 g. This is a direct analogy to the fact that .1g /d=2 2 L1,1 for the Laplace–Beltrami operator g on a d -dimensional closed Riemannian manifold. If an unbounded Fredholm module has dimension d then we show that ResD is a vector-valued linear form on all operators of order d in Connes’ quantum calculus, and that, if ! is an extended limit on l1 (or, equivalently, a state on the algebra l1 =c0 ) then the formula Tr! .A/ D !.ResD .A// holds for any Dixmier trace applied to an operator A of order d in the calculus.

383

Section 12.1 Introduction

Example 10.3.7 in Chapter 10, and Section 11.7 in the last chapter, suggest that the functional Int! .A/ :D Tr! .AhDid / D !.ResD .AhDid //,

A 2 L.H /

(12.1)

should be viewed as the integral in Connes’ quantum calculus. We have to use distinct terminology for those A 2 L.H / such that the value Int! .A/ is independent of the choice of !. Definition 12.1.1. Let .H , D/ be an unbounded Fredholm module of dimension d , that is, hDid 2 L1,1 . (a) If A 2 L.H / we say the value Int! .A/ as above, where ! is an extended limit on l1 , is the noncommutative integral of A. (b) If the value Int! .A/ does not depend on the choice of extended limit !, then we say A is q-measurable and denote the value Int.A/. Due to Lemma 9.7.4, the notion of q-measurability of the operator A coincides with the Dixmier measurability of the operator AhDid , where the notion of Dixmier measurability was defined in Definition 9.1.1. We investigate several aspects of the noncommutative integral, which, essentially, is the final investigation of the book. Assume that the spectrum of the operator D satisfies the following Weyl asymptotic condition, jn jd n,

n  0.

This is not a standard imposition in noncommutative geometry, although common and frequently used examples of unbounded Fredholm modules satisfy this condition. The Weyl asymptotic condition is a very natural one given the intention that D 2 be a noncommutative version of the Hodge–Laplacian g . If we use the following notation for the logarithmic means M : l1 ! l1 , Pn .M.x//n :D



1 j C1 , 1 j D0 j C1

j D0 xj

Pn

x 2 l1 , n  0,

then the noncommutative integral (12.1) has the following form in terms of convergence of logarithmic means of expectation values. Tauberian results can be used to obtain conditions when the noncommutative integral of a bounded operator is, in fact, the limit of its expectation values. The following theorem is the main result of this chapter.

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Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.1.2. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl asymptotic condition as above, and let fej g1 j D0 be an ordered eigenbasis for the operator D. (a) For any extended limit ! on l1 the noncommutative integral Int! defined in equation (12.1) is a singular state on the algebra L.H /, and the noncommutative integral of A 2 L.H / is given by an extended limit applied to the logarithmic mean of the expectation values of A, Int! .A/ D .! ı M/.fhAen , en ig1 nD0 /.

(12.2)

(b) A bounded operator A 2 L.H / is q-measurable if and only if its expectation values logarithmically converge, and then Int.A/ D .lim ıM/.fhAen , en ig1 nD0 /. (c) If A 2 L.H / is q-measurable and the expectation values of A 2 L.H / have slow oscillation at infinity, meaning that lim .hAen , eni  hAem , em i/ D 0 when n > m ! 1 such that log log n  log log m ! 0, then Int.A/ D lim hAen , en i. n!1

Another result we show in this chapter says that the noncommutative integral Int! is a hypertrace on a maximal C  -algebra associated to trace class perturbations. Definition 12.1.3. A state ' on L.H / is a hypertrace on a C  -algebra N  L.H / if '.AB/ D '.BA/ for all A 2 N and B 2 L.H /. Theorem 12.1.4. Let .H , D/ be an unbounded Fredholm module of dimension d . Then N :D fA 2 L.H / : ŒA, hDid  2 L1,1 \ .M1,1 /0 g is the maximal C  -subalgebra of L.H / such that Int! is a hypertrace on N for every dilation invariant extended limit ! on l1 . Here .M1,1 /0 is the ideal obtained by closing the trace class operators L1 in the symmetric norm k  kM1,1 . Theorem 12.1.2 and Theorem 12.1.4 are contained, and proven, in Section 12.3. We remark in that section on the notion of quantum ergodicity, which is the principle that the limit of the sequence of expectation values exists and provides a “measure” (in the sense of Theorem 12.1.2 (c)). We also remark in Section 12.3 on how Theorem 12.1.4

Section 12.2 The Noncommutative Residue in Noncommutative Geometry

385

implies the commonly known result that if ŒD, A is bounded for A belonging to a norm dense subset of a C  -algebra N  L.H /, then Int! is a hypertrace on N for all extended limits on l1 . In Sections 12.4 and 12.5 we look at a few simple examples of the noncommutative residue in noncommutative geometry. We show that the noncommutative residue is an invariant of isospectral deformation. In the examples of Connes, Landi and DubiosViolette, the noncommutative residue of the isospectral deformation of a pseudo-differential operator of order d can be calculated by the noncommutative residue of the pseudo-differential operator itself. Thus, for isospectral deformations, which presently are the main class of examples of noncommutative geometries, the Dixmier trace of operators in the quantum calculus and the quantum integral are calculated directly by symbols and the classical Lebesgue integral. We illustrate by considering the noncommutative torus, which is an isospectral deformation of smooth functions on the ordinary torus that is related to Weyl quantization. Finally, in Section 12.6, the last section of the book, we touch on some future applications of singular traces. The heat kernel formulation of Dixmier traces is associated to partition functions, or Gibbs states. The expectation value formulas mentioned above are associated to increasing to infinity energy levels of quantum systems, or the notion of large quantum numbers. We reflect on how the coincidence of these formulations with singular traces, which has been derived in the different sections of this book, give a fundamental, and interesting, mathematical formalization of the link between high temperature limits as classical limits, the correspondence principle in quantum mechanics, and the notion of singular traces.

12.2 The Noncommutative Residue in Noncommutative Geometry Chapter 11 introduced the vector-valued noncommutative residue of a Hodge–Laplacian modulated operator T : L2 .X/ ! L2 .X) on a closed Riemannian manifold X,  Res.T / D

1

n X 1 hT ej , ej i log.2 C n/ nD0 j D0

where Œ denotes an equivalence class in l1 =c0 and fej g1 j D0 is an ordered eigenbasis of the Hodge–Laplacian g . When the Hodge–Laplacian modulated operator T is localized (see Definition 11.6.10), so that it locally has a modulated symbol, this spectral formula reduces to an asymptotic integral formula on the cotangent bundle. In particular, Wodzicki’s noncommutative residue on classical pseudo-differential operators of order d becomes a special case of the spectral formula. This opens the field for an analogous notion of noncommutative residue in spectral extensions of differential geometry, such as Alain Connes’ noncommutative geometry.

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Chapter 12 Residues and Integrals in Noncommutative Geometry

Connes’ calculus is based on spectral triples, .A, H , D/. Here A is an  -algebra of bounded operators on the Hilbert space H . Unlike the commutative algebra of functions C 1 .X/ on a closed Riemannian manifold X, where functions act by pointwise multiplication on the Hilbert space L2.X/, the algebra A can be noncommutative. The operator D : Dom.D/ ! H is self-adjoint with compact resolvent. It is intended to be the analog of a first order Dirac operator (mainly inspired by the Hodge–Dirac operator or the Dirac operator of a spin manifold [146]), and the positive operator D 2 is the analog of the Hodge– Laplacian. The “analogy” comes in the form of replicating the algebraic and spectral properties of the operators of differential geometry, as below. Predominately we are interested in the data .H , D/, which, when singled out, is called an unbounded Fredholm module. Given an unbounded Fredholm module .H , D/, the functional calculus of selfadjoint operators allows us to define the positive operator with trivial kernel, hDis :D .1 C D 2 /s=2 ,

s 2 R.

When s < 0 this operator (extends to) a compact operator on H . Sobolev spaces can be generalized by the Connes–Moscovici approach. Set H s :D Dom.hDis /,

s > 0,

with norm kxkH s :D khDis xk,

x 2 H s.

Let op m , m 2 R, denote the set of continuous linear operators A : H s ! H sm , 8s 2 R. Then op m contains the noncommutative versions of pseudo-differential operators [54], (compare also with Proposition 10.3.4). An unbounded Fredholm module .H , D/ is said to have dimension d > 0 if hDid 2 L1,1 . Henceforth we assume that all unbounded Fredholm modules have dimension d . The analogy of the graded algebra formed by op m , m 2 R, to the pseudo-differential operator calculus is given in the next lemma. Lemma 12.2.1. Let .H , D/ be an unbounded Fredholm module of dimension d as above. (a) If A 2 op 0 then A is a bounded operator on H .

Section 12.2 The Noncommutative Residue in Noncommutative Geometry

387

(b) If A 2 op m , m < 0, then A is a compact operator, and if A 2 op m , m < d , then A is a trace class operator. (c) If A 2 op d then A 2 L1,1 . Proof. Suppose A 2 op 0 . Since H 0 H then A : H ! H is continuous and linear. This proves (a). If A belongs to op m , m 2 R, then note that AhDim and hDim A belong to op 0 . In which case there exists B 2 op 0 such that A D BhDim . Since hDim is compact, m < 0, it follows that A 2 op m is compact. Since hDid 2 L1,1 then A 2 op d belongs to L1,1 . This proves (c). If m < d then hDim D .hDid /r , where r D m.d /1 > 1. By Lemma 8.6.2, hDim 2 L1 and it follows that A 2 op m , m < d , is trace class. There is a wider class of operators than those in op d that we will deal with simultaneously. The following operators are the analog of the Hodge–Laplacian modulated operators introduced in Chapter 11. Definition 12.2.2. Let .H , D/ be an unbounded Fredholm module of dimension d . Then A 2 L.H / is .H , D/-modulated (equivalently, hDid -modulated in the sense of Definition 11.2.1) if kAkmod :D sup t 1=2 kA.1 C t hDid /1 k2 < 1. t >0

The definition, while useful in an abstract sense, is not always convenient to test. Section 11.3 noted that there was a more convenient formula, involving symbols, by which to define a Laplacian modulated operator. In noncommutative geometry there are no symbols, the replacements are spectral formulas. An orthonormal basis of fej g1 j D0 is called an ordered eigenbasis of D if Dej D j ej , j  0, where j0 j j1 j    are the eigenvalues of D order by increasing absolute value (note this is possible since D has compact resolvent). The eigenvalues of D will always appear ordered by increasing absolute value. When a Weyl asymptotic condition is satisfied by the eigenvalues (as in the statement below) the modulated condition is more conveniently tested by the action of a bounded operator A on the ordered eigenbasis fej g1 j D0 . Compare the next lemma with Lemma 11.3.13. Lemma 12.2.3. Let .H , D/ be an unbounded Fredholm module of dimension d satisfying the Weyl condition (i.e. hn id n, n  0), and let fej g1 j D0 be an ordered eigenbasis of D as above. Then A 2 L.H / is .H , D/-modulated if and only if 1 X j DnC1

kAej k2 D O..1 C n/1 /,

n  0.

388

Chapter 12 Residues and Integrals in Noncommutative Geometry

Proof. Fix n  0 and let Pn be the projection onto the linear span of vectors ej , j  n C 1. Then kAPn k22 D

1 X

kAej k2 .

j DnC1

Since hDid 2 L1,1 satisfies the Weyl condition, it follows that there exist constants c2 > c1 > 0 such that



  c1 c2 Pn EhDid 0, . EhDid 0, nC1 nC1 The assertion now follows from Lemma 11.2.5. We summarize the properties that can be found in Section 11.2 concerning modulated operators, noting that, by assumption, 0 hDid 2 L1,1 . Lemma 12.2.4. Let .H , D/ be an unbounded Fredholm module of dimension d , and let fej g1 j D0 be an ordered eigenbasis of D as above. (a) If A 2 L.H / is .H , D/-modulated, then A, diag.fhAen , en ig1 nD0 / 2 L1,1 . (b) If A 2 L.H / is .H , D/-modulated, then n X

hAej , ej i D O.log.2 C n//,

n  0.

j D0

(c) If A 2 L.H /, then AhDid is .H , D/-modulated. (d) If A 2 op d then A is .H , D/-modulated. Proof. The assertion of (a) is directly shown by Theorem 11.2.3. The assertion of (b) follows by arguing as in the proof of Lemma 11.6.5. The assertion of (c) follows from Proposition 11.2.2. The assertion of (d) follows from (c) since AhDid 2 L.H / if A 2 op d . Part (b) of the lemma shows that we can define a vector-valued noncommutative residue which is a linear form on operators that are .H , D/-modulated. Part (d) says that the residue is a linear form on operators of order d in the quantum calculus. Definition 12.2.5. Let .H , D/ be an unbounded Fredholm module of dimension d , and let fej g1 j D0 be an ordered eigenbasis of D as above. Define the noncommutative

Section 12.2 The Noncommutative Residue in Noncommutative Geometry

389

residue associated to .H , D/ by  ResD .A/ :D

1

n X 1 , hAej , ej i log.2 C n/ nD0

A 2 L1,1 ,

j D0

where Œ denotes an equivalence class in l1 =c0 . If the equivalence class ResD .A/ contains a constant sequence, then we say that ResD .A/ is scalar valued. The bijection between scalars and equivalence classes of constant sequences will be implicit. The main result on modulated operators (Theorem 11.2.3) is central to connecting the noncommutative residue associated to an unbounded Fredholm module to the established use of the Dixmier trace in noncommutative geometry (NCG). The connection places the noncommutative residue at the heart of the integration half of Connes’ quantized calculus. Since the proof of the next theorem is identical to that of Corollary 11.2.4 we omit it. Recall from Lemma 9.7.4 that every extended limit ! on l1 can be used to define a Dixmier trace on the ideal L1,1 . Recall also that a trace on L1,1 is normalized if it takes the value 1 on the operator diag.f.n C 1/1 g1 nD0 /. Theorem 12.2.6 (Trace Theorem NCG). Let .H , D/ be an unbounded Fredholm module of dimension d . Then every .H , D/-modulated operator A belongs to L1,1 and (a) for a Dixmier trace Tr! , where ! is an extended limit on l1 , Tr! .A/ D !.ResD .A//. (b) A is Dixmier measurable (takes the same value for every Dixmier trace) if and only if ResD .A/ is scalar valued (and then Tr! .A/ D ResD .A/, for every extended limit ! on l1 ). (c) '.A/ D ResD .A/ for a scalar value ResD .A/ for all normalized traces ' : L1,1 ! C if and only if n X hAej , ej i D ResD .A/  log.n/ C O .1/ , n  1, j D0

for a scalar value ResD .A/, where feng1 nD0 is an ordered eigenbasis of D.

390

Chapter 12 Residues and Integrals in Noncommutative Geometry

Part (a) of the theorem indicates that Dixmier traces are the only traces whose value on .H , D/-modulated operators can be obtained by combining the noncommutative residue with an extended limit. We will not discuss further in this chapter the stronger version of measurability given in Part (c) of the theorem. It could be argued, given the results in Chapter 11, that considering this stronger version of measurability is warranted. However, we will use the “traditional” form of the scalar noncommutative residue based on Dixmier traces. We restrict ourselves to the comment that one could use the set of all normalized traces on L1,1 instead of the set of Dixmier traces and make adjusted statements in what follows. Remark 12.2.7. The convenience of the noncommutative residue is that it is a unique class. The disadvantage is that it is vector-valued. The reduction of the vector-value to a unique scalar value is not always possible (not all .H , D/-modulated operators are measurable). Connes’ notion of measurability [48], as studied in Chapter 9 of this book, is revealed as precisely the condition for ResD to be scalar valued.

12.3 The Integral in Noncommutative Geometry If g : C 1 .X/ ! C 1 .X/ is the Laplace–Beltrami operator on a closed d -dimensional Reimannian manifold X, then Example 10.3.7 in Chapter 10, and Section 11.7 in the last chapter, showed that the functional A ! Tr! .A.1  g /d=2 / D !.Res.A.1  g /d=2 // provides, when A D Mf , f 2 L2 .X/, integration of square integrable functions on X. It should not be overlooked that the same functional, when A is a pseudodifferential operator of order 0, is a trace on the  -algebra of pseudo-differential operators of order 0. Thus, “commutative” geometry, which is sometimes how differential geometry is referred to within noncommutative geometry, contains the precedent of the above functional providing a trace on a noncommutative algebra. If .H , D/ is an arbitrary unbounded Fredholm module of dimension d , the observations above suggest that the functional Int! .A/ :D Tr! .AhDid / D !.ResD .AhDid //,

A 2 L.H /,

(12.3)

be viewed as the integral in Connes’ quantum calculus. For convenient normalization we assume that Int! .1/ D 1, or, what is the same condition (see Theorem 9.7.5), n X 1 hj id D 1, n!1 log.2 C n/

Tr! .hDid / D ResD .hDid / D lim

j D0

for every extended limit ! on l1 . In this section we examine the spectral properties of the integral in (12.3), and its association with the logarithmic means of expecta-

Section 12.3 The Integral in Noncommutative Geometry

391

tion values of the operator A 2 L.H /. This “quantum ergodic” property is a spectral equivalent of the Lebesgue integral. We will also show that Int! , in analogy to its differential geometry prototype, is a trace on the algebra of operators of order 0 in the Connes–Moscovici quantum calculus. We recall from Definition 12.1.1 that q-measurable operators are those operators A 2 L.H / such that the value Int! .A/ is independent of the choice of !. In that case we write Int.A/ for the unique value. Since essentially bounded functions and classical zero order pseudo-differential operators are examples of q-measurable operators in differential geometry, it is natural in noncommutative geometry to “impose” the condition of q-measurability on the algebra A in a spectral triple .A, H , D/. The next theorem characterizes what is being imposed. Assuming Weyl asymptotics, it shows that q-measurability of a bounded operator is equivalent to logarithmic convergence of its expectation values. Theorem 12.3.1. Let .H , D/ be an unbounded Fredholm module of dimension d . Suppose that the operator D satisfies the Weyl asymptotic condition, hn id n, n  0. Let feng1 nD0 be an eigenbasis of D such that Den D nen where j0 j j1 j    are increasing. (a) If A 2 L.H /, then, for every extended limit ! on l1 , we have Int! .A/ D .! ı M/.fhAen , en ig1 nD0 /. (b) An operator A 2 L.H / is q-measurable if and only if the sequence fhAen , en ig1 nD0 of expectation values logarithmically converge, and then Int.A/ D .lim ıM/.fhAen , en ig1 nD0 /. Proof. Let A 2 L.H /. By Lemma 12.2.4 (c) the operator AhDid is .H , D/-modulated, so Theorem 12.2.6 (a) provides the formula   n X 1 d hAhDi ej , ej i Int! .A/ D ! log.2 C n/ j D0   n X 1 d hj i hAej , ej i . D! log.2 C n/ j D0

By the Weyl asymptotic condition, we have hj id D .1 C o.1//=.j C 1/, j  0. Hence, n n X X 1 hAej , ej i C o.log.n//. hj id hAej , ej i D j C1 j D0

j D0

392

Chapter 12 Residues and Integrals in Noncommutative Geometry

Since ! is an extended limit, then   Pn hAej ,ej i  n X 1 hAej , ej i j D0 1Cj . Int! .A/ D ! D! Pn 1 log.2 C n/ 1Cj j D0 1Cj 

j D0

The last equality follows by applying Lemma 6.3.5, since Pn lim

n!1

1 j D0 1Cj

log.2 C n/

! 1C .

The assertion of (a) is shown. The assertion of (b) follows from (a) and Lemma 9.3.6. The observations in Theorem 12.3.1 bring the noncommutative integral into the realm of the classical theory of divergent sequences. Tauberian theorems for the logarithmic mean can be used to determine if the sequence of expectation values converges. The ordinary limit calculates the noncommutative integral in this case. We give an example of a Tauberian condition, the condition of slow oscillation at infinity [142], and also [163]. Corollary 12.3.2. Let .H , D/ be an unbounded Fredholm module of dimension d satisfying the Weyl asymptotic condition, and let feng1 nD0 be an ordered eigenbasis of D. If A is q-measurable and the expectation values of A 2 L.H / have slow oscillation at infinity, meaning that lim .hAen , en i  hAem , em i/ D 0 when n > m ! 1 such that log log n  log log m ! 0, then Int.A/ D lim hAen , en i. n!1

Proof. Let sn :D hAen , en i and s D fsn g1 nD0 . If A is q-measurable then s is logarithmically convergent. Using Kwee’s Tauberian result [142, Lemma 3], and the remark [163, Remark 3], the condition of slow oscillation at infinity of s implies that s is convergent. The logarithmic means are regular, so if s is convergent then .lim ıM/.s/ D lim s and the assertion is shown. Despite us imposing the condition that the operator D satisfies the Weyl asymptotic condition, hn id n, n  0, which implies that hDid is Dixmier measurable in the sense of Chapter 9, we note that there are many operators A 2 L.H / which are not q-measurable. The expectation A ! fhAen , en ig1 nD0 , L.H / ! l1 ,

393

Section 12.3 The Integral in Noncommutative Geometry

is onto, so it suffices, to know that there are bounded operators that are not q-measurable, to know only that there are bounded sequences that are not logarithmically convergent. Example 12.3.3. Suppose that the Weyl asymptotic condition is satisfied and A 2 L.H / is such that hAen , en i D

1 X j D0

Œ222j ,222j C1 / .n/,

n  0.

Then A is not q-measurable. P 1 Proof. Set xn :D 1 j D0 Œ222j ,222j C1 / .n/, n  0, and x D fxn gnD0 . Since kxk1 1, then x 2 l1 . We show that M.x/ has convergent subsequences which converge to different values, hence M.x/ 62 c. Consider the subsequence 22n

M.x/.2

22n

22mC1

2X n1 X 2X 1 xk 1 1 /D C o.1/ D C o.1/ 2n 2n k C 1 log.22 / kD0 k C 1 log.22 / mD0 2m 2 kD2 n1 X

n1 X

D

1 1 22m log.2 / C o.1/ D 22m log.2/ C o.1/ 2n 2n 2 log.2/ log.22 / mD0 mD0

D

1 C o.1/, 3

n  0.

Similarly, M.x/.22

2nC1

/D

2 C o.1/, 3

n  0.

It follows that x is not logarithmically convergent. We now consider under what conditions the noncommutative integral Int! is a hypertrace on a C  -subalgebra N  L.H /. Hypertraces were defined in Definition 12.1.3. Theorem 12.3.4. Let .H , D/ be an unbounded Fredholm module of dimension d (a Weyl asymptotic condition on D is not assumed). Then N :D fA 2 L.H / : ŒA, hDid  2 L1,1 \ .M1,1 /0 g is the maximal C  -subalgebra of L.H / such that Int! is a hypertrace on N for every extended limit ! on l1 . Here .M1,1 /0 is the ideal obtained by closing the trace class operators in the symmetric norm k  kM1,1 .

394

Chapter 12 Residues and Integrals in Noncommutative Geometry

Proof. First, we show that N is indeed a C  -algebra. We omit showing that N is a  -algebra, as these properties are an easy calculation using the fact that L1,1 \ .M1,1 /0 is a two-sided ideal. If T 2 L1,1 , recall the definition of the quasi-norm kT k1,w :D supn0 .1 C n/1 .n, T /. Evidently, kATBk1,w kAk1 kT k1,w kBk1 for all A, B 2 L.H /. Let An ! A as n ! 1 in the uniform topology where An 2 N . Then A 2 L.H / satisfies kŒhDid , A  ŒhDid , An k1,w 4kA  An k1 khDid k1,w ! 0. By assumption ŒhDid , An  2 L1,1 \ .M1,1 /0 , n  0, hence ŒhDid , A 2 L1,1 \ .M1,1 /0 since L1,1 \ .M1,1 /0 is closed in the quasi-norm topology. It follows that A 2 N. Observe that Int! .ŒA, B/ D Tr! .ABhDid  BAhDid / D Tr! .BhDid A  BAhDid / D Tr! .BŒhDid , A/,

A, B 2 L.H /

(12.4)

by using the trace property of a Dixmier trace Tr! . Now suppose that ŒhDid , A 2 L1,1 \ .M1,1 /0 . Since L1,1 \ .M1,1 /0 is an ideal, it follows that BŒhDid , A 2 L1,1 \ .M1,1 /0 ,

B 2 L.H /.

Hence, we have that Tr! .BŒhDid , A/ D 0 since all Dixmier traces, as continuous traces on M1,1 , vanish on L1,1 \ .M1,1 /0 . From (12.4), it follows that Int! is a hypertrace on N for every extended limit !. Conversely, if Int! is a hypertrace on N for every extended limit ! then, from (12.4), Tr! .BŒhDid , A/ D 0 for all Dixmier traces and every B 2 L.H /. In particular, choose B such that BŒhDid , A D jŒhDid , Aj. So Tr! .jŒhDid , Aj/ D 0 for all Dixmier traces. We recall from Theorem 10.1.3 (or Theorem 9.2.1) that this condition implies that ŒhDid , A 2 L1,1 \ .M1,1 /0 . Corollary 12.3.5. Let .H , D/ be an unbounded Fredholm module of dimension d . For any extended limit ! on l1 , (a) Int! is a hypertrace on the C  -algebra N :D fA 2 L.H / : ŒD, A 2 L.H /g.

395

Section 12.3 The Integral in Noncommutative Geometry

(b) Int! is a hypertrace on the C  -algebra N :D fA 2 L.H / : ŒhDi, A 2 L.H /g. Proof. Let d D kr , k 2 N and k > d . Then 0 < r < 1 and ŒhDid , A D hDid ŒhDid , AhDid X  k d r .ki / r r .i 1/ hDid D hDi hDi ŒhDi , AhDi i D1

D

k X

hDii r ŒhDir , AhDi.kC1i /r .

i D1

Suppose that ŒD, A is bounded. It follows (see [42, 248] or [186, Theorem 2.4.3]) that ŒhDir , A is bounded. The individual terms in the sum satisfy, for 1 i k, hDii r ŒhDir , AhDi.kC1i /r 2 L d ,1 L ir

If B1 2 L d ,1 and B2 2 L ir

d .kC1i/r ,1

.n, B1 / const  .n C 1/i r=d ,

d .kC1i/r ,1

.

, then

.n, B2 / const  .n C 1/.i k1/r=d ,

n  0.

Hence, Z

Z

1

kB1B2 k1

1

.s, B1 /.s, B2 /ds const  0

.s C 1/.kC1/r=d ds < 1.

0

Hence ŒhDid , A is trace class. By Theorem 12.3.4 the state Int! is a hypertrace for the  -algebra of operators A 2 L.H / such that ŒD, A is bounded. It follows from Theorem 12.3.4 that Int! is a hypertrace for the operator norm closure of this  -algebra. This proves the assertion of (a). The assertion of (b) follows from (a) (as applied to the operator hDi instead of D). Remark 12.3.6. Alain Connes and Henri Moscovici [54], introduced zero order noncommutative pseudo-differential operators as those A 2 op 0 such that the operators k k ıhDi .A/, k  0, are bounded where ıhDi denotes the k th power of the derivation ıhDi .A/ :D ŒhDi, A. We can then observe that, by Corollary 12.3.5 (b), the noncommutative integral Int! is a hypertrace on the zero order operators in the Connes– Moscovici pseudo-differential calculus.

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Chapter 12 Residues and Integrals in Noncommutative Geometry

12.4 Example of Isospectral Deformations The most commonly studied noncommutative geometries lie within the category of isospectral deformations of commutative geometries. This section shows that the noncommutative residue is an invariant of isospectral deformation. Let X be a closed d -dimensional Riemannian manifold where d  2, with associated unbounded Fredholm module .L2.X/,

p

g /

where g : C 1 .X/ ! C 1 .X/ is the Laplace–Beltrami operator associated to a fixed metric g (see Section 10.3 and Section 11.6). We scale the Laplace–Beltrami operator, as given in Definition 10.3.3, by the factor 

Vol.X/  Vol.Sd 1 / d.2/d

2=d

1  D 4



Vol.X/ .d=2 C 1/

2=d

so that Tr! ..1  g /d=2 / D 1. Usually, the Fredholm module would be formed from the pair of the Hodge–Dirac operator and the Hilbert space of square integrable sections of the exterior bundle. However, to consider p the noncommutative residue it is sufficient to consider the module above. The .L2.X/, g /-modulated operators, in the terminology of Section 12.2, are the Hodge–Laplacian modulated operators p introduced in Section 11.6, and the noncommutative residue associated to .L2.X/, g / is the residue on Hodge–Laplacian operators introduced in Definition 11.6.6. Let Iso.X/ denote the compact Lie group of isometries of the closed manifold X [133, 167]. We recall that an isometry ˛ : X ! X is a diffeomorphism which preserves the metric g. The topology on Iso.X/ is the compact-open topology. The group Iso.X/ acts on the Hilbert space L2.X/ by the unitaries .V˛ f /.x/ :D f .˛ 1 .x//,

f 2 L2 .X/, ˛ 2 Iso.X/.

The unitary V˛ commutes with the Laplacian for any ˛ 2 Iso.X/, Œg , V˛  D 0. This commutation property is the defining feature of an isometry amongst other diffeomorphisms [111, 250]. Suppose that we have a smooth action by isometries of the r -torus Tr , r  2, on the manifold X (equivalently a smooth embedding of Tr in Iso.X/ such that the map .s, x/ ! ˛s .x/, s 2 Tr , x 2 X, is smooth in s and x). Denote .Vs f /.x/ :D f .˛s .x//,

f 2 L2 .X/, s 2 Tr .

397

Section 12.4 Example of Isospectral Deformations

Using this action, we define an isospectral deformation. The deformation involves the discrete version of the well-known Moyal product [95, 192]. Define, Z e 2 i hm,siVs ds, m 2 Zr (12.5) Um :D Tr

where the integral converges in the strong sense since Vs is a strongly continuous unitary representation of Tr on L2 .X/ (shown below in Lemma 12.4.5). We also show below that the operators Um , m 2 Zr , form a complete system of pairwise orthogonal projections. Let ƒ denote the additive group of skew symmetric matrices in Mr .T/, and suppose that ‚ 2 ƒ. If A 2 L.L2.X//, then define the isospectral deformation of A, formally, by X e i  hn,‚.m/iUm AUn . (12.6) L‚ .A/ :D m,n2Zr

To associate this formal definition to a linear operator L‚.A/, the convergence of the square partial sums X e i  hn,‚.m/iUm AUn , k  0, (12.7) m,n2Œk,kr

needs to be considered. The partial sums are bounded operators, but they may not converge in the uniform or strong sense when ‚ 6D 0. The following theorems are the main results of this section. They state that the noncommutative residue is an invariant of isospectral deformation, and, further, that the isospectral deformation of Hodge– Laplacian modulated operators is a complete trace invariant. Theorem 12.4.1. If A 2 L.L2 .X// is a Hodge–Laplacian modulated operator, then L‚.A/ 2 L.L2 .X// is a Hodge–Laplacian modulated operator (where L‚.A/ is the L2 -limit of the partial sums in (12.7) in the ideal of Hilbert–Schmidt operators), and Res.L‚ .A// D Res.A/,

8‚ 2 ƒ.

In particular, if A : C 1 .X/ ! C 1 .X/ is a classical pseudo-differential operator of order d , then Res.L‚ .A// D ResW .A/,

8‚ 2 ƒ

where ResW is Wodzicki’s noncommutative residue on the closed Riemannian manifold X. The proof of the theorem is given below. First we show that a stronger statement about trace invariance is true. If g is the Laplace–Beltrami operator, we recall from Section 11.6 that 0 .1g /d=2 2 L1,1 and that any Hodge–Laplacian modulated operator belongs to the ideal L1,1 .

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Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.4.2. If A 2 L.L2 .X// is a Hodge–Laplacian modulated operator, then A, L‚.A/ 2 L1,1 and '.L‚ .A// D '.A/,

8‚ 2 ƒ

for every trace ' : L1,1 ! C. In particular, Tr! .L‚.A// D Tr! .A/ 8‚ 2 ƒ, for every Dixmier trace Tr! , and L‚ .A/ is Dixmier measurable if and only if A is Dixmier measurable. This result allows us to calculate the noncommutative integral of the deformation L‚.Mf /, Mf 2 L.L2.X//, of a smooth function f 2 C 1 .X/. Note that we shorten L‚.Mf / to L‚.f /. Then L0 .f / D Mf , f 2 C 1 .X/. Fix the matrix ‚ 2 ƒ. The following characterization of the isospectral deformation of a smooth function is the Peter–Weyl form [53]. Theorem 12.4.3. If f 2 C 1 .X/ then L‚ .f / 2 L.L2 .X// where L‚.f / is the strong limit of the partial sums in (12.7). (a) Equivalently, L‚ .f / D

X m2Zr

MUm f V 1 ‚.m/ , 2

where L‚ .f / is the uniform limit of the partial sums of the right-hand side of the display. (b) The operator L‚ .f / is a q-measurable operator with Z 1  Int.L‚.f // D f .u/du. Vol.X/ X We observe, as a corollary to Theorem 12.4.2, that L‚.A/ can be associated to a closeable unbounded operator for any bounded operator A 2 L.L2 .X//, and that the formula (12.3) can be used to extend the noncommutative integral to unbounded operators of the form L‚.A/ . Corollary 12.4.4. If A 2 L.L2 .X//, then L‚ .A/ defines a continuous linear operator L‚.A/ : H d .X/ ! L2.X/, where H d .X/ denotes the Sobolev space of order d described in Section 10.3 (L‚.A/ is the strong limit of the square partial sums on H d .X/). The operator L‚ .A/.1  g /d=2 has a bounded extension as a Hodge–Laplacian modulated operator, and defining Int! on L‚ .A/ by the same formula as in (12.3), we have Int! .L‚.A// D Int! .A/.

399

Section 12.4 Example of Isospectral Deformations

We now prove the results. Lemma 12.4.5. The action s ! Vs , s 2 Tr , is strongly continuous. Proof. It is sufficient to show continuity as s ! 0, and on a connected component of X. Suppose Xi is a connected component. We claim that k.Vs  1/f k2 ! 0 as s ! 0 for every function f 2 C 1 .Xi /. We have k.Vs  1/f k2 Vol.Xi /  k.Vs  1/f k1 kf kLip  sup dist.x, ˛s .x//. x2Xi

Then k.Vs  1/f k2 ! 0 as s ! 0 since ˛s is smooth. The claim is shown. Now let fn 2 C 1 .Xi / converge to f 2 L2 .Xi / as n ! 1 in the L2-norm. Then lim k.Vs  1/f k2 lim k.Vs  1/fn k2 C 2kfn  f k2 D 2kfn  f k2.

s!0

s!0

Since n can be arbitrarily large, it follows that s ! Vs is strongly continuous on L2.Xi /. Since X D [N i D1 Xi is a finite disjoint union of connected components, then N L2.X/ D ˚i D1 L2.Xi / and it follows that s ! Vs is strongly continuous on L2.X/. The preceding lemma justifies the definition of the operators Um given in (12.5). Lemma 12.4.6. The operators Um , m 2 Zr , are pairwise orthogonal projections and P the series m2Zr Um converges to 1 strongly. Proof. It follows from (12.5) that, for m, n 2 Zr , Z Z 2 i .hn,t iChm,si/ e Vs Vt dsdt D Um Un D T2r

T2r

e 2 i .hn,t iChm,si/VsCt dsdt .

Using the substitution t ! t  s, Z e 2 i .hn,t iChmn,si/Vt dsdt Um Un D T2r

Z D

Tr

e

2 i hmn,si

 Z ds 

Tr

e

2 i hn,t i

 Vt dt

D ı0.m  n/Un .

Similarly, we have Un D

Z Tr

e 2 i hm,si Vs ds D

Z Tr

e 2 i hm,uiVu du D Un .

Hence, the operators Um , m 2 Zr , are pairwise orthogonal projections.

400

Chapter 12 Residues and Integrals in Noncommutative Geometry

P We now prove that the series m2Zr Um converges to 1 strongly. The Hilbert space L2.X/ can be decomposed into a direct sum of eigenspaces for g . Since the unitaries Vs , s 2 Tr , commute with g , each of these eigenspaces is Vs -invariant, s 2 Tr . Since each of these eigenspaces is finite dimensional, there exists a common eigenbasis en, n  0, for Vs , s 2 Tr . For every n  0, the r -torus acts on the 1-dimensional space spanned by en according to the formula Vs en D e 2 i hm,si en ,

s 2 Tr ,

for some m 2 Zr (m depends on n). It is now clear from (12.5) that Um en D en . Since en, n  0, form an orthonormal basis of L2 .X/, the assertion follows. Lemma 12.4.7. Let pk , k  0, be pairwise orthogonal projections such that P 1 kD0 pk D 1. If j˛k,l j D 1 for k, l  0, then the mapping A!

1 X

˛k,l pk Apl ,

A 2 L2

k,lD0

is an isometry of L2 into itself. Here, the series converges in L2. Proof. Fix a common eigenbasis en, n  0, for the projections pk , k  0. Choose sets Ak such that pk en D en for n 2 Ak and pn ek D 0 otherwise. For A 2 L2 and anm :D hAen , em i, we have that kAk22 D

1 X n,mD0

janm j2 D

1 X

X

k,lD0 n2Ak ,m2Al

janm j2 D

1 X

kpk Apl k22 .

k,lD0

is an orthogonal decomposition of the Hilbert space In other words, fpk L2 pl g1 k,lD0 L2. The assertion follows. We now show that, in the case where A is Hilbert–Schmidt, the series in the formal definition (12.6) converges to a Hilbert–Schmidt operator. Fix a matrix ‚ 2 ƒ. Corollary 12.4.8. For every A 2 L2 , we have L‚.A/ 2 L2 and kAk2 D kL‚.A/k2 . The series (12.6) converges in L2 . Proof. The proof follows from a combination of (12.5), Lemma 12.4.7 and Lemma 12.4.6. Using the last lemma we can show, in the case where A is a Hodge–Laplacian modulated operator, that the square partial sums in the formal definition (12.6) converge in L2 to a Hodge–Laplacian modulated operator.

401

Section 12.4 Example of Isospectral Deformations

Lemma 12.4.9. If A 2 L2 is Hodge–Laplacian modulated, then L‚.A/ is Hodge– Laplacian modulated and kL‚.A/kmod D kAkmod, where L‚.A/ is the Hilbert– Schmidt operator from Corollary 12.4.8. Proof. Let A be Hodge-Laplacian modulated. From Section 11.2 we know that A is Hilbert–Schmidt and so, by Corollary 12.4.8, L‚.A/ is Hilbert–Schmidt. For brevity, denote Wt :D .1  t .1  g /d=2 /1 2 L.L2.X//, t > 0. Note that ŒWt , Vs  D 0, s 2 Tr , t > 0, since Œg , Vs  D 0, s 2 Tr . Hence, taking into account that Wt also commutes with Um , m 2 Z2 , we have that L‚.A/Wt D L‚ .AWt / 2 L2. By Corollary 12.4.8 again kL‚.A/Wt k2 D kL‚.AWt /k2 D kAWt k2 Hence, kL‚.A/kmod D sup t 1=2 kL‚.A/Wt k2 D sup t 1=2 kAWt k2 D kAkmod. t >0

t >0

We prove the main results. Proof of Theorem 12.4.1. By assumption the operator A 2 L.L2.X// is Hodge– Laplacian modulated, and then Lemma 12.4.9 proves that L‚.A/ is Hodge–Laplacian modulated where L‚.A/ is the Hilbert–Schmidt operator from Corollary 12.4.8. The first assertion of Theorem 12.4.1 follows. Recall that the unitaries Vs , s 2 Tr , and hence the projections Um , m 2 Zr , commute with g . Select a common eigenbasis en , n  0, for g and Um , m 2 Zr , where g en D 2nen and n , n  0, are increasing. Let pN , N  0, be the projection onto the linear span of fe0, : : : , eN g. Since pN is a finite projection, we have X e  i hn,‚.m/i Tr.Um AUn pN /. Tr.L‚ .A/pN / D m,n2Zr

By construction, pN commutes with the (pairwise orthogonal projections) Um , m 2 Zr . Hence, Tr.Um AUn pN / D 0 for m ¤ n. Therefore, X Tr.L‚ .A/pN / D Tr.AUn pN / D Tr.ApN / since that

n2Zr

P n2Zr

Un converges strongly to 1 and Tr is strongly continuous. This proves

1

1 Tr.L‚.A/pN / Res.L‚.A// D log.2 C Tr.pN // N D0 1

 1 Tr.ApN / D Res.A/. D log.2 C Tr.pN // N D0 

402

Chapter 12 Residues and Integrals in Noncommutative Geometry

The assertions in Theorem 12.4.1 are proved. The particular statement when A is a classical pseudo-differential operator of order d now follows from Corollary 11.6.15.

Proof of Theorem 12.4.2. By assumption the operator A is V -modulated where V :D .1g /d=2 . It follows from Lemma 12.4.9 that the operator L‚ .A/ is V -modulated. By Theorem 11.2.3, we have that A and L‚ .A/ belong to L1,1 and, by the choice of basis, that N X

.k, L‚ .A// D Tr.L‚.A/pN / C O.1/,

kD0

N X

.k, A/ D Tr.ApN / C O.1/,

kD0

where pN is the projection in the proof of Theorem 12.4.1 above. Recall from the above proof that Tr.L‚.A/pN / D Tr.ApN /. Now, by Lemma 5.7.5, we have that N X

.k, L‚ .A/  A/ D O.1/.

kD0

It follows from Theorem 5.7.6 that L‚ .A/  A 2 Com.L1,1 /. All the assertions in the theorem now follow. We now prove Theorem 12.4.3. Lemma 12.4.10. For f 2 C 1 .X/, we have that X kUm f k1 < 1. m2Zr

Proof. Define the function h : Tr  X ! C by setting h.s, x/ :D f .˛s .x//. By the assumptions on f and the assumption of a smooth action of the torus on X, h is a smooth function on Tr  X. Select 0 ¤ m D .m1, : : : , mr / 2 Zr and let jmk j be the maximal term amongst jm1 j, : : : , jmr j. Using (12.5) together with integration by parts, we obtain Z .Um f /.x/ D e 2 i hm,sih.s, x/ds Tr Z @r C1 D .2 imk /r 1 e 2 i hm,si r C1 h.s, x/ds. @sk Tr Therefore, for some constant depending on the function f , we have kUm f k1 .2jmk j/r 1  khkC rC1.Tr X/ const  hmir 1 . The statement follows since hmir 1 , m 2 Zr , is summable.

403

Section 12.4 Example of Isospectral Deformations

Lemma 12.4.11. For every f 2 L1 .X/ and g 2 L2 .X/, we have .Um Mf Un /g D .Un g/.Umn f /,

m, n 2 Zr .

Proof. For every g 2 L2.X/, we have from (12.5) that, for m, n 2 Zr , Z .Um Mf Un /g D

T2r

e 2 i .hm,siChn,t i/Vs Mf Vt gdsdt .

However, for s, t 2 Tr , Vs Mf Vt g D .Vs Mf /.g ı ˛t / D Vs .f  .g ı ˛t // D .Vs f /.Vs .g ı ˛t // D .Vs f /.Vt Cs g/ and, therefore, Z .Um Mf Un /g D

Z

T2r

e 2 i .hmn,siChn,t Csi/.Vs f /.Vt Cs g/dsdt

e 2 i .hmn,siChn,ui/.Vs f /.Vu g/dsdu Z  Z  D e 2 i hmn,si Vs f ds e 2 i hn,uiVugdu D

T2r

Tr

Tr

D .Umn f /.Un g/. Proof of Theorem 12.4.3. Fix k, l  0 and ‚ 2 ƒ. It follows from (12.5) and a simple computation that, for m 2 Zr , X

e  i hn,‚.m/iUn D

jnjl

X

 Un V‚.m/=2 .

jnjl

Consider the following partial sums (we apply below Lemma 12.4.11). X

e  i hn,‚.m/i .Um Mf Un /g D

jmnjk,jnjl

X

e  i hn,‚.m/i .Um f /.Un g/

jmjk,jnjl

D

X jmjk

D

X jmjk

X   i hn,‚.m/i .Um f / e Un g jnjl

  X  .Um f / Un V‚.m/=2 g . jnjl

404

Chapter 12 Residues and Integrals in Noncommutative Geometry

When l ! 1, the latter sum converges (in the norm topology) in L2 .X/ to the function X bk :D .Um f /.V‚.m/=2 g/, k 2 N. jmjk

When k ! 1, the latter sum converges uniformly in L2 .X/ by Lemma 12.4.10. Indeed, X X .U f /.V g/ kUm f k1 kV‚.m/=2gk2 ! 0, m ‚.m/=2 2

k1 0, 1

Tr.Ae ˇ D / Gˇ .A/ :D D 1 2 Tr.e ˇ D / 2

P1

ˇ 1 2n nD0 hAen , enie P1 ˇ 12 n nD0 e

,

A 2 L.H /.

The terms P1

1

j D0 e

ˇ 1 2j

e ˇ

12 n

,

n  0,

represent the probability of being in the eigenstate en , n  0. What happens to these states as ˇ ! 1 is called the high temperature limit [116]. Conceptually, as ˇ ! 1 the probability of being in any eigenstate is becoming uniform. With higher energy, all eigenstates are becoming saturated and, essentially, the quantum states are becoming indistinguishable. The states Gˇ , ˇ > 0, may not have a limit as ˇ ! 1. A limit can be obtained in an extended sense by applying an extended limit on L1 .RC /. Theorem 12.6.1 shows that for certain extended limits, the limit of the Gibbs states is the noncommutative integral

413

Section 12.6 Classical Limits

in noncommutative geometry, and that an operator A 2 L.H / is q-measurable if the states Gˇ .A/, ˇ > 0, logarithmically converge as ˇ ! 1. The expectation A ! fhAen , en ig1 nD0 , L.H / ! l1 , associates to a self-adjoint operator A, representing an observable, the expected observable value for the eigenstate en , n  0. The correspondence principle in quantum mechanics is that the “classical limit” is the large quantum number limit of the quantum system, that is, the limit as n ! 1 when feng1 nD0 is an ordered eigenbasis. The quantum harmonic oscillator is an example of the principle [244]. Conceptually, the “limit at infinity” of the sequence fhAen , en ig1 nD0 represents the observable value in the classical limit. The sequence, generally, does not converge. A limit can be obtained in an extended sense by applying an extended limit on l1 . Theorem 12.3.1 shows that, for extended limits composed with the logarithmic mean, the limit of the expectation values is the noncommutative integral in noncommutative geometry. The next theorem mathematically unites the “correspondence principle” and the “high temperature limit” to each other and to Connes’ noncommutative integral. We restrict an extended limit ! on L1 .0, 1/ to an extended limit on l1 by !.x/ :D !

X 1

 xn Œn,nC1/ ,

x D fxng1 nD0 2 l1 .

nD0

Recall that an orthonormal basis feng1 nD0 is an ordered eigenbasis of D if Den D nen , n  0, and jn j, n  0, is an increasing sequence. Recall also that an unbounded Fredholm module .H , D/ satisfies the Weyl asymptotic condition if jn jd n, n  0. In the statement below M denotes the discrete logarithmic means (see the definition before Theorem 12.1.2) and M denotes the continuous logarithmic means (see the definition before Theorem 9.2.1). Theorem 12.6.1. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl asymptotic condition. If A 2 L.H / then Int! .A/ D !

ı M.fhAej , ej ig1 j D0 /

1 2  Tr.Ae ˇ D / , D!ıM 1 2 Tr.e ˇ D /



for any ordered eigenbasis feng1 nD0 of D. Here ! is an extended dilation invariant limit on L1 .0, 1/ such that ! ı log is also dilation invariant, and the two formulas on the left involve the restriction of ! to an extended limit on l1 . Connes’ notion of measurability in noncommutative geometry coincides with the sequence of expectation values and the Gibbs states converging logarithmically.

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Chapter 12 Residues and Integrals in Noncommutative Geometry

Theorem 12.6.2. Let .H , D/ be an unbounded Fredholm module satisfying the Weyl asymptotic condition and feng1 nD0 be an ordered eigenbasis of D. An operator A 2 L.H / is q-measurable if and only if the following values exist and coincide Int.A/ D

lim ıM.fhAej , ej ig1 j D0 /

1 2  Tr.Ae ˇ D / . D lim ıM 1 2 Tr.e ˇ D /



Theorem 12.6.1 and Theorem 12.6.2 are proved below. First we require a preliminary lemma. Lemma 12.6.3. If 0 A 2 L1,1 is such that .n, A/ .1 C n/1 , n  0, then   1  q1 ˇ 1 Aq / D .1 C o.1// 1 C ˇ Tr.e , ˇ ! 1, q > 0. (12.10) q Proof. Fix  > 0 and select n  1 sufficiently large so that k.k, A/ 2 .1  , 1 C / for k  n. Then, 1 X

e ˇ

1

.1/q k q

C O.1/ Tr.e ˇ

1

Aq

/

kD1

1 X

1

.1C/q k q

C O.1/.

kD1

Therefore, Z Z 1 1 q q 1 q e ˇ .1/ s dsCO.1/ Tr.e ˇ A / 0

e ˇ

1

e ˇ

1

.1C/q s q

dsCO.1/.

0

Computing the integrals in terms of the -function, we obtain that     1 1 1=q ˇ 1 Aq 1=q / .1 C /ˇ  1 C C O.1/ Tr.e C O.1/. .1  /ˇ  1 C q q Since  is arbitrarily small, the assertion follows. Lemma 12.6.4. Let 0 A 2 M1,1 be positive and Dixmier measurable and let B 2 L.H /. The operator AB is Dixmier measurable if and only if the following limit exists for some q > 0,   1 .ˇA/q / .t /. Tr.Be lim M t !1 ˇ Proof. We use results and terminology from Chapters 6, 7, 8 and 9. Set  D Tr. Suppose that the limit exists. Setting f .t / :D exp.t q /, t > 0, and fixed q > 0, we infer that the heat kernel functionals !,B,f .A/ take the same value for every dilation invariant extended limit !. It follows from Corollary 8.5.2 that ! .AB/ does

415

Section 12.6 Classical Limits

not depend on !. By Theorem 8.3.6 and Theorem 6.4.1, the set of Dixmier traces and the set of heat kernel formulas coincide. Hence, ! .AB/ does not depend on ! and, therefore, AB is Dixmier measurable. Conversely, let AB be Dixmier measurable. Since A is self-adjoint, it follows that ! .A.0,

(12.13)

was suggested as a replacement dimension condition for infinite dimensional triples [46, 47], and surfaces in quantum field theory applications of noncommutative geometry [47, §7], [117], [48, IV.9]. An alternative integral based upon Gibbs states was proposed for the condition (12.13) at ([37], p. 208) and in ([94], §2.1.3) (for brevity we use the same notation as the referenced texts):  1 2  Tr.Ae r D / , A 2 N, (12.14) A :D ` Tr.e r 1 D2 / `

where ` was an unspecified limiting procedure at infinity. That the functional in (12.14) yields the same value as a Dixmier trace for a finite dimensional triple was not shown in ([37], p. 208) or ([94], §2.1.3) outside the assumption that A is q-measurable. Hardy, in [108, §3.8], illustrated the difference between the Cesàro and logarithmic means. In particular, the logarithmic means are less trivial (more useful) than the Cesàro means, meaning that morePsequences converge logarithmically than converge arithmetically. If p D pn diverges, Hardy defined the means .N, p/ : l1 ! l1 , fpn g1 nD0 > 0 and Pn j D0 xj pj ..N, p/.x//n :D Pn , x D fxj g1 j D0 2 l1 , n  0. p j D0 j The CesàroPmeans are the .N, 1/ means and the logarithmic means are the .N, .1 C n/1/ means. As pn diverges more quickly the means .N, p/ becomes more trivial, until reaching the point that only convergent sequences end up being .N, p/-convergent [108, Theorem 15]. An exact statement for when a logarithmically convergent sequence is a convergent sequence is known [163, Theorem 5]: a logarithmically convergent sequence x D fxn g1 nD0 2 l1 is a convergent sequence if and only if ˇ ˇ Œnr  X ˇ 1 xk  xn ˇˇ ˇ lim lim sup ˇ r ˇ D 0. k r!1C n!1 .Œn   n/ log.1 C n/ kDnC1

418

Chapter 12 Residues and Integrals in Noncommutative Geometry

Here Œnr  denotes the ceiling of nr , n  1, r > 1. The more testable Tauberian criteria of slowly oscillating real sequences was shown by B. Kwee in [142]. Equivalent criteria to slowly oscillating is given in [163]. Observations between the noncommutative integral and the logarithmic mean of expectation values were first noted in [154]. For the notion of quantum ergodicity see [43,74,204,220,266]. The expectation value approach to traces has also been used in Pietsch’ work on traces on Banach ideals [184], as well as in [126], which extended [154]. Connes [47,48] first noted the hypertrace property of the noncommutative integral for spectral triples. The proof we have given follows [42] and observations in [33]. Isospectral Deformations The Myers–Steenrod Theorem, [167], proves that the isometry group Iso.X / of a compact manifold X is a compact Lie group. The Lie algebra of Iso.X /, i.e. the infinitesimal generators of isometries, is composed of the Killing vector fields on X . If the dimension of X is d , as a finite dimensional smooth manifold the dimension of Iso.X / is less than or equal to d.d C 1/=2 [134]. From the theory of compact Lie groups, the maximal connected, commutative, subgroups of a compact Lie group all have the same dimension r , and are all isomorphic to the r -torus [219]. These subgroups are called maximal torii and the value r is called the rank of the compact Lie group. Therefore, closed manifolds whose isometry group Iso.X / has rank  2 admit smooth representations of the torus, and are available for isospectral deformation [53]. Killing fields relate to metric preserving symmetries of the manifold. Two central Killing fields are enough to guarantee a rank  2 for the isometry group [219, Corollary 5.13]. Conversely, if it is known that the compact Lie group Tr acts smoothly on a compact manifold X , then there is a metric g on X such that Tr  Iso.X / [202, Corollary 6.3]. That is, one can always choose a metric g such that the smooth action of a torus Tr can be represented by unitaries that commute with the Laplace–Beltrami operator g . Connes, Landi, and Dubios-Violette [52,53] introduced the notion of a noncommutative geometry associated to an isospectral deformation of a closed Riemannian manifold. It involves the discrete version of the Moyal product [101, 164, 192]. Bruno Iochum, Victor Gayral and their collaborators [95, 96], considered the continuous Moyal product and isospectral deformations of non-compact manifolds. Their paper [96] showed the result in Corollary 12.4.3 for compact (or, under certain conditions, non-compact) Riemannian manifolds. That if A is Hilbert–Schmidt, then the deformation L‚ .A/ is Hilbert–Schmidt was shown in [96], as was the property that Tr.L‚ .A/p/ D Tr.Ap/ for a spectral projection p of the Hodge– Laplacian. The method of proof of Theorem 12.4.3 in Section 12.4 is different from that in [96], which was based on -function estimates. -function estimates cannot show that, as in Theorem 12.4.2, isospectral deformation of modulated operators is trace invariant for every trace on L1,1 . The exposition of isospectral deformations given in Sections 12.4 and 12.5 is close to [261]. The noncommutative torus originated, as an example of a noncommutative geometry, with [44,48,50,55] and [246]. It appears now in innumerable papers, particularly in connection with String Theory, e.g. [51, 137]. The noncommutative torus has other, equivalent, presentations besides as an isospectral deformation. Usually the emphasis is on irrational values for the parameter  [191]. Connes used a -function residue to recover the trace on the noncommutative torus in [47]. The computation of the -function of the noncommutative torus can be found in [87].

Section 12.7 Notes

419

Classical Limits For deeper aspects of Connes’ quantum calculus, such as the local-index formula, and recovery of the Einstein–Hilbert action, see [30, 31, 48, 50, 54, 131]. Connes’ posited the Dixmier trace as a replacement for taking the classical limit in [45]. There is no completely definitive approach to taking the “classical limit”. The high temperature limit is an approach in statistical mechanics [116], the correspondence principle goes back to the time of Bohr, as does the Ehrenfest notion of expected values [16, 145], mathematically there is strict quantization of Poisson structures [144]. The interesting conceptual feature of the invariance of the noncommutative integral under isospectral deformation is that it, the noncommutative integral, is not so much a classical limit, but rather the stronger property of being an invariant of quantization.

Appendix A

Operator Results

A.1

Matrix Results

The following results on matrices were used at specific places in the text, especially for the results on quasi-nilpotent compact operators in Chapter 5. In Section 5.4, if u : C ! R is continuous we defined a function uO : Mn .C/ ! R by setting X u.A/ O :D u./, A 2 Mn .C/. 2 .A/

Set logC.x/ :D maxflog.jxj/, 0g,

x 2 R.

b

b

Lemma A.1.1. For every operator A 2 Mn .C/, we have logC .A/ logC.jAj/. Proof. Let N1 :D jf 2 .A/ : jj > 1gj, N2 :D jf 2 .jAj/ : jj > 1gj. It follows from Lemma 1.1.20 that Y

N1 Y

jj D

2 .A/,jj>1

j.k, A/j

kD0

N1 Y

.k, A/.

kD0

If N2  N1 , then .k, A/ > 1 for every k 2 .N1 , N2 . Therefore, N1 Y

.k, A/

kD0

N2 Y

Y

.k, A/ D

kD0

jj.

(A.1)

2 .jAj/,jj>1

If N2 < N1 , then .k, A/ 1 for every k 2 .N2 , N1 . Therefore, N1 Y kD0

.k, A/

N2 Y kD0

Y

.k, A/ D

2 .jAj/,jj>1

Combining (A.1) and (A.2), we conclude the proof.

jj.

(A.2)

421

Section A.1 Matrix Results

In Section 2.3 we defined the distribution function of an operator L.H / (or a matrix), nA .s/ :D Tr.EjAj .s, 1//,

s  0.

Lemma A.1.2. For every normal operator A 2 Mn .C/ and every ˛ 2 C such that j˛j 1, we have ˇ ˇ X X ˇ ˇ ˇ ˇ nA .1/.   ˛  ˇ ˇ 2 .˛A/,jj>1

Proof. It is clear that ˇ X ˇ ˇ ˛ ˇ 2 .˛A/,jj>1

2 .A/,jj>1

X 2 .A/,jj>1

ˇ ˇ D ˇˇ

ˇ ˇ ˇˇ

X 2 .A/,11

ˇ ˇ 1ˇˇ D nA .1/.

Lemma A.1.3. Let A1 , A2 , A3 2 Mn .C/ be normal operators such that A1 C A2 C A3 D 0. We have ˇ 3 ˇX ˇ ˇ

X

i D1 2 .Ai /,jj>1

ˇ ˇ ˇˇ 2.nA1 .1/ C nA2 .1/ C nA3 .1//.

Proof. Let pi D EjAi j .1, 1/, i D 1, 2, 3. Since every Ai is normal, it follows that pi D EAi .fz 2 C, jzj > 1g/. Hence, X

 D Tr.pi Ai pi /,

8i D 1, 2, 3.

2 .Ai /,jj>1

Set

_

pD

pi .

i D1,2,3

We have Tr.pi Ai pi / D Tr.pAi p/  Tr..p  pi /Ai .p  pi //. Since A1 C A2 C A3 D 0, it follows that 3 X i D1

Tr.pi Ai pi / D 

3 X i D1

Tr..p  pi /Ai .p  pi //.

422

Appendix A Operator Results

We have k.p  pi /Ai .p  pi /k1 k.1  pi /Ai .1  pi /k1 1 and rank..p  pi /Ai .p  pi // Tr.p  pi / D Tr.p/  Tr.pi /. It follows from the obvious inequality jTr.A/j kAk1  rank.A/ that Tr..p  pi /Ai .p  pi // Tr.p/  Tr.pi /. Therefore,

ˇX ˇ 3 ˇ ˇ

X

i D1 2 .Ai /,jj>1

ˇ X 3 ˇ ˇˇ Tr.p/  Tr.pi /. i D1

It is clear that Tr.pi / D nAi .1/ for i D 1, 2, 3 and Tr.p/ nA1 .1/ C nA2 .1/ C nA3 .1/. The assertion follows immediately.

A.2

Operator Inequalities

In this section .M, / denotes a von Neumann algebra M equipped with a fixed normal semifinite trace . The results presented below can be found in [18, 23, 32]. Lemma A.2.1. Let A, C 2 M be self-adjoint operators such that A  C . It follows that .AC /  .CC/. Proof. There exists projection p 2 M such that CC D pCp. We have p.A  C /p  0 and, therefore, pAp  CC. Hence, pAC p  pA p C CC. Thus, .CC/ .pA p C CC/ .pAC p/ .AC /. Lemma A.2.2. Let A, B 2 M be positive operators. We have .B 1=2 .A  1/C B 1=2 /  ..B 1=2 AB 1=2  1/C/ provided that B 1. Proof. Let C D 1 C .A  1/C . We have A C and, therefore, B 1=2 AB 1=2  1 B 1=2 CB 1=2  1.

423

Section A.2 Operator Inequalities

It follows from Lemma A.2.1 that ..B 1=2 AB 1=2  1/C/ ..B 1=2 CB 1=2  1/C /. We have B 1 and, therefore, B 1=2 CB 1=2  1 B 1=2 .C  1/B 1=2 . It follows from Lemma A.2.1 that ..B 1=2 CB 1=2  1/C / .B 1=2 .C  1/B 1=2 / D .B 1=2 .A  1/CB 1=2 /. Lemma A.2.3. Let A, B 2 M be positive operators. For every convex continuous increasing function f , we have .B 1=2 f .A/B 1=2 /  .f .B 1=2 AB 1=2 // provided that B 1. Proof. First, consider the case when supp.A/ 2 L1 .M, /. We have .supp.B 1=2 AB 1=2 //, .supp.B 1=2 f .A/B 1=2 // .supp.A//. On the interval Œ0, kAk1 , the function f admits a uniform approximation with piecewise-linear convex increasing continuous functions. Hence, for every  > 0, there exists n X g:t! ˛k .t  ˇk /C , kD1

such that ˛k , ˇk  0, 1 k n, and f .t / g.t / f .t / C ,

80 t kAk1 .

By Lemma A.2.2, we have .B 1=2 g.A/B 1=2 /  .g.B 1=2 AB 1=2 //. It follows that .B 1=2 f .A/B 1=2 /  .f .B 1=2 AB 1=2 //    .supp.A//. Since  > 0 is arbitrarily small, the assertion is proved for every operator A with supp.A/ 2 L1 .M, /.

424

Appendix A Operator Results

In the general case, we have      1 1=2 1=2 1=2 1=2 .B f .A/B / D lim  B f AEA ,1 B n!1 n      1 1=2 1=2 D .f .B 1=2 AB 1=2 //.  lim  f B AEA ,1 B n!1 n Lemma A.2.4. Let A, B 2 M be positive operators. We have B 1=2 .A  1/CB 1=2 .B 1=2 AB 1=2  1/C provided that B  1 and .A/  f0g [ .1, 1/. Proof. Let t > .supp.A//. There exists  > 0 such that .t  , A/ D 0. It follows that .t , B 1=2 AB 1=2 / ., B/  .t  , A/ D 0. If t < .supp.A//, then .t , A/  1 and, therefore 1 .t , A/ kB 1 k1  .t , B 1=2 AB 1=2 / .t , B 1=2 AB 1=2 /. It follows that .B 1=2 AB 1=2 /  f0g [ .1, 1/.

(A.3)

.B 1=2 supp.A/B 1=2 /  f0g [ .1, 1/.

(A.4)

Similarly,

We have ker.B 1=2 AB 1=2 / D f 2 H : B 1=2 AB 1=2  D 0g D B 1=2 ker.A/. Similarly, ker.B 1=2 supp.A/B 1=2 / D B 1=2 ker.A/. Therefore, supp.B 1=2 AB 1=2 / D supp.B 1=2 supp.A/B 1=2 /.

(A.5)

It follows from (A.4) and (A.5) that supp.B 1=2 AB 1=2 / B 1=2 supp.A/B 1=2 .

(A.6)

It follows from (A.3) that .B 1=2 AB 1=2  1/C D B 1=2 AB 1=2  supp.B 1=2 AB 1=2 /.

(A.7)

425

Section A.2 Operator Inequalities

On the other hand, we have B 1=2 .A  1/CB 1=2 D B 1=2 AB 1=2  B 1=2 supp.A/B 1=2 .

(A.8)

The assertion follows from (A.6), (A.7) and (A.8). Lemma A.2.5. Let A, B 2 M be positive operators. We have .B 1=2 .A  1/CB 1=2 / ..B 1=2 AB 1=2  1/C/ provided that B  1. Proof. Let C D AEA .1, 1/. We have A  C and, therefore, B 1=2 AB 1=2  1  B 1=2 CB 1=2  1. It follows from Lemma A.2.1 that ..B 1=2 AB 1=2  1/C/  ..B 1=2 CB 1=2  1/C /. It follows from Lemma A.2.4 that .B 1=2 CB 1=2  1/C  B 1=2 .C  1/CB 1=2 D B 1=2 .A  1/C B 1=2 . The assertion follows immediately. Lemma A.2.6. Let A, B 2 M be positive operators. For every convex continuous increasing function f , we have .B 1=2 f .A/B 1=2 / .f .B 1=2 AB 1=2 // provided that B  1. Proof. First, consider the case when supp.A/ is finite. We have .supp.B 1=2 AB 1=2 //, .supp.B 1=2 f .A/B 1=2 // .supp.A//. On the interval Œ0, kAk1  kBk1 , the function f admits a uniform approximation with piecewise-linear convex increasing continuous functions. Hence, for every  > 0, there exists n X g:t! ˛k .t  ˇk /C , kD1

such that ˛k , ˇk  0, 1 k n, and f .t / g.t / f .t / C ,

80 t kAk1  kBk1 .

426

Appendix A Operator Results

By Lemma A.2.5, we have .B 1=2 g.A/B 1=2 / .g.B 1=2 AB 1=2 //. It follows that .B 1=2 f .A/B 1=2 / .f .B 1=2 AB 1=2 // C   .supp.A//. Since  > 0 is arbitrarily small, the assertion is proved for finitely supported operator A. In the general case, we have      1 1=2 1=2 1=2 1=2 ,1 B .B f .A/B / D lim  B f AEA n!1 n      1 lim  f B 1=2 AEA D .f .B 1=2 AB 1=2 //. , 1 B 1=2 n!1 n The following theorem is an immediate corollary of Lemma A.2.3 and Lemma A.2.6. Theorem A.2.7. Let A, B 2 M be positive operators and let f be a convex continuous function such that f .0/ D 0. We have (a) .B 1=2 f .A/B 1=2 /  .f .B 1=2 AB 1=2 // if B 1. (b) .B 1=2 f .A/B 1=2 / .f .B 1=2 AB 1=2 // if B  1. Lemma A.2.8. Let A, C 2 M be positive operators. We have (a) .1 C A/1  C.1 C CAC /1 C if C 1. (b) .1 C A/1 C.1 C CAC /1 C if C  1. Proof. We prove only the first assertion. Let Cn :D maxfC , 1=ng, n  1. It follows that 1 C Cn ACn  Cn.1 C A/Cn . Hence, .1 C Cn ACn /1 Cn1.1 C A/1 Cn1 . It follows that .1 C A/1  Cn .1 C Cn ACn /1 Cn ! C.1 C CAC /1 C . If f is a power function, one can prove a significantly stronger variant of Theorem A.2.7.

427

Section A.2 Operator Inequalities

Lemma A.2.9. Let A, B 2 M be positive operators and let s > 0. We have (a) .B 1=2 AB 1=2 /1Cs B 1=2 A1Cs B 1=2 if 0 B 1. (b) .B 1=2 AB 1=2 /1Cs  B 1=2 A1Cs B 1=2 if B  1. Proof. For brevity, we set C :D B 1=2 . We use the following formula for the power of a positive operator D, Z sin.s/ 1 s t D.1 C tD/1 dt . Ds D  0 It follows that CA1Cs C D

sin.s/ 

and 1Cs

.CAC /

sin.s/ D 

Z

Z

1

t s CA.1 C tA/1 AC dt .

0

1

t s CAC.1 C t CAC /1 CAC dt .

0

It follows that  .CA1Cs C  .CAC /1Cs / D sin.s/ Z

1

D

t s CA..1 C tA/1  C.1 C t CAC /1 C /AC dt .

0

The assertion now follows from Lemma A.2.8.

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Index

Amplitude 319 Basis (of a Hilbert space) Eigen see Eigenbasis Orthonormal 16, 26 Bicommutant 39 Theorem 39 Bounded operator 15 Absolute value 19 Adjoint 16 Hermitian see Bounded operator, Selfadjoint Normal 19, 41 Positive 16, 39 Self-adjoint 16 Unitary 26 C  -algebra 39 Maximal such that the noncommutative integral is a hypertrace 393 Calculus Connes’ quantum 303, 306, 386, 395, 417 Functional 19 Pseudo-differential 323, 326, 333 Calkin Algebra 35 Correspondence see Calkin correspondence Function space 53 Operator space 53, 56 Sequence space 23, 53, 153 Calkin correspondence 22, 23, 27, 53, 54, 57, 71, 75, 79, 117, 152 Caratheodory theorem 82 Cesàro operator Continuous 120, 278, 281 Discrete 154, 184

Classical Differential geometry see Riemannian Geometry Pseudo-differential operator see Pseudo-differential operator, Classical Classical limit 385, 411, 413, 419 Closed commutator subspace 154, 162 Closed Riemannian manifold 332, 363, 376, see also Riemannian geometry Isometry Group 396, 418 Commutant 38 Commutator subspace 154, 155, 182, 190, 304, 380 Closed see Closed commutator subspace History 192 Of the weak-l1 ideal 191 Compact operator 17 -compact 53, 64 Decomposition into normal and quasinilpotent 22, 155, 173, 189 Finite rank 16 Nilpotent 20, 173 Normal 19, 22, 155, 162, 173, 188 Quasi-nilpotent 20, 168, 173, 188 Compactly based operator 326, 350 Compactly supported operator 326, 350 Continuous trace see Trace, Continuous Correspondence Calkin see Calkin correspondence Principle in quantum mechanics 413 Cosphere bundle see Riemannian geometry Decomposition Jordan see Jordan decomposition

446 Of a compact operator into normal and quasi-nilpotent parts see Compact operator, Decomposition into normal and quasi-nilpotent Of a self-adjoint operator into positive and negative parts 42, 157 Of a symmetric functional into hermitian parts 110 Of a symmetric functional into normal and singular parts 108, 123 Of an operator into real and imaginary parts 42, 157 Polar 48 Schmidt 49 Yosida-Hewitt 68, 70, 72 Dilation Operator 85, 197 Semigroup 197 Distribution 318 Spectral see Spectral, Distribution function Tempered 318 Dixmier -Macaev ideal see Ideal, DixmierMacaev Measurable see Measurable operator, Dixmier Letter to the Luminy conference see Dixmier, on the origin of Dixmier traces On the origin of Dixmier traces 217 Dixmier trace 29, 77, 194, 221, 276, 296, 313, 316, 390, 411 Calculation of using -function residue formulas 222, 245, 267, 313 Calculation of using eigenvalues see Lidskii formula of Calculation of using expectation values 241, 307, 340, 361, 371, 389 Calculation of using heat kernel formulas 222, 245, 251, 313, 414 Connes’ influence on 152, 218, 300 Continuous version of 199 Discrete version of 195, 205, 214 Distribution formulas for 226, 233, 236

Index

Invariant under isospectral deformation 398 Lidskii formula of 221, 232, 242, 314, 360 Linearity of 30 Link between discrete and continuous versions 205 Link to fully symmetric functionals 78, 195, 203, 222, 312 Link to the noncommutative residue 304, 360, 370, 389 Non-normality of 45 On a fully symmetric operator ideal 78, 206, 208, 213 On a Lorentz function space 200, 226 On a Lorentz ideal 198, 199, 221, 226, 236 On a Lorentz operator space 218 Wodzicki construction of 195, 216 Wodzicki representation, theorem 216 Eigenbasis Of the Hodge-Laplacian see Laplacian, Eigenbasis Ordered 340, 363, 384, 387, 413 Eigenprojection see Spectral, Projection Eigenvalue 17 (Algebraic) multiplicity 18 Geometric multiplicity 17 Matrix results on 420 Of a Hodge-Laplacian modulated operator 367 Of a Laplacian modulated operator 355 Eigenvalue sequence 18, 77, 153, 173, 182, 232, 297, 312, 340, 355, 366 Eigenvalue theorem 355, 367 Expectation operator 101, 117 Expectation values 307, 340, 358, 365, 384, 391, 413 Exponential map 287, 293 Extended limit 196, 222 Applied to the noncommutative residue 305, 360 At 0 219 C -invariant 281, 291

Index

Composed with the logarithmic mean 235, 391, 413 Dilation and power invariant 267, 413 Dilation invariant 30, 197, 228, 245, 250, 384 M -invariant 221, 235, 245, 272, 293 On sequences 32, 196, 316 On unbounded measurable functions 206, 227 On unbounded positive sequences 206 Fekete lemma 92 Figiel-Kalton theorem 108, 120, 152, 154 Closed version 163 Fredholm module see Unbounded Fredholm Module Fubini’s theorem, a special form 171 Fully symmetric Function space 59, 130 Ideal of compact operators 194, 206 Operator space 59, 142 Trace see Trace, Fully symmetric With Fatou norm 59, 133, 143, 146 Fully symmetric functional 69, 107, 130, 142 Approximation of by Dixmier traces see Dixmier Trace, On a fully symmetric operator ideal Distinction from symmetric functional 76, 109, 133 Existence of 108, 130, 142 Relatively normal 210, 211 Functional Fully Symmetric see Fully symmetric functional Minkowski see Minkowski functional Normal 68 Singular 67 Symmetric see Symmetric functional Unitarily invariant see Trace Geometric stability see Ideal, Geometrically stable Gibbs state 412, 417 Hahn-Banach theorem 123 Invariant version 197

447 Harmonic function 168 Heat kernel functional 222, 244, 246, 276, 313 Generalised 245, 256, 259 Link to classical limit and Gibbs states 414 Link to fully symmetric functionals 245, 255 High temperature limit 412 Hilbert space 15, 153, 194, 339, 382, 386, 412 Of square integrable functions 40, 303, 331, 363, 372, 396, 405 Hodge-Laplacian modulated see Modulated operator, Hodge Laplacian Hypertrace 384, 393 Ideal L1,1 (weak-l1) 26, 187, 297, 303, 315, 359 M1,1 26, 29, 203, 221, 235, 297, 303, 311 M1,1.M, / 222, 244, 246, 272, 276 Arithmetically mean closed 72 Banach see Symmetric ideal Banach sum 63 Dixmier-Macaev (compact operators) see Ideal, M1,1 Dixmier-Macaev (general) see Ideal, M1,1.M, / Geometrically stable 173, 180, 193 Lorentz see Lorentz, Ideal Of a semifinite von Neumann algebra 55 Of finite rank operators 24 Of nuclear operators see Trace class operators Of trace class operators see Trace class operators Orlicz 186 Schatten(-von Neumann) 24, 61, 63, 185, 192 Symmetric see Symmetric ideal Symmetrically normed see Symmetric ideal Two-sided of compact operators 22, 56 Weak-lp 24, 185

448 Integral As a faithful normal semifinite trace 44 Noncommutative see Noncommutative Integral Of functions (recovery of the Lebesgue integral) 305, 329, 334, 338, 372, 379, 406 On a manifold 331, 376, 398 Isometry group Of a closed manifold see Closed Riemannian manifold, Isometry group Isospectral deformation 385, 397 Of a smooth function 398, 408 Of the torus see Noncommutative Torus Tracial invariance of 398, 408 Jordan decomposition Of operator 232 Of symmetric functional 109 Karamata theorem 279 For dilation invariant extended limits 266 Köthe dual see Lorentz operator space Kwapien theorem 122 Laplace-Beltrami operator see Laplacian, Hodge Laplacian Eigenbasis of 306, 338, 356, 363, 406 Eigenvalues of see Weyl, Law Hodge 306, 332, 363, 396 Modulated see Modulated operator, Laplacian On Euclidean space 304, 318, 321, 325, 336, 349 On the torus 405 Lidskii formula 33, 77, 153, 183, 312, 316 Of a Dixmier trace see Dixmier trace, Lidskii formula Of the canonical trace (Lidskii’s Formula) 185 Lifting Kalton’s of subharmonic functions 168

Index

Lemma 175 Of symmetric function or sequence spaces to symmetric operator spaces 97 Of symmetric functionals on function spaces to operator spaces 75, 107, 117, 142, 200 Of symmetric sequence spaces to symmetric function spaces 101 Limit Classical see Classical limit Extended see Extended limit Linear extension 27 Logarithmic mean 221, 235, 244, 273, 383, 413 Of expectation values 308, 383, 391, 413 Lorentz Function space 65, 72, 199, 226 Ideal of compact operators 25, 66, 194, 198, 232 Operator ideal 66, 200, 221, 273 Operator space 65, 67, 200 Marcinkiewicz see Lorentz Change in terminology 72 Mean Cesàro see Cesàro operator Difference between Cesàro and Logarithmic 417 Logarithmic see Logarithmic mean Measurable operator Dixmier (or in Connes’ sense) 223, 272, 276, 299, 313, 340, 360, 371, 383, 389, 390 Formulas for 314, 414 Link between Dixmier and M - 223, 272, 290 Link between Dixmier and Tauberian 299, 314, 316 M - 223, 272, 287 q- 383, 391, 414 Stronger version 390 Tauberian 297, 314 Minkowski functional 90, 95 Modulated function 345, 374

Index

Modulated operator 336, 339 .H , D/- 387 Hodge-Laplacian 338, 363, 377, 385, 397 Isospectral deformation of 397 Laplacian 304, 336, 349, 360, 373 Localised 367 Pseudo-differential operator as an example of 351, 364 Set of as a bimodule for the pseudodifferential operators of order 0 364 Mollifier 318 Moyal product 397, 418 Multi-index 317 Von Neumann algebra 38 -finite 43, 49 Atomic 40, 45, 58, 69, 79, 108, 118, 143, 153, 194, 246, 272 Atomless 40, 45, 52, 58, 69, 79, 108, 118, 143, 218, 240, 246, 272 Bimodule of 55, 68 Center of 39, 55 Commutative 39 Factor 39, 55, 68, 69, 153 Finite 43 Lattice of projections see Projection, Lattice Maximal commutative 39 Of bounded operators on a separable Hilbert space 15, 38, 108, 153, 194 Of essentially bounded functions 40, 44, 63, 407 Pre-dual 62 Semifinite 43, 68 Nigel Kalton Origin of modulated operator 380 Origin of the description of the commutator subspace of the trace class operators 192 Origin of uniform submajorization 104 Non-measurable operator 314 Non q-measurable 392 Pseudo-differential 306, 362, 371

449 Noncommutative Lp ideals see Ideal, Schatten(-von Neumann) Lp spaces 53, 60 Integration see Trace, Normal and see Trace, Semifinite Integration à la Connes 68, 70, see also Noncommutative Integral Measure theory see Von Neumman algebra, Semifinite Pseudo-differential operators see Calculus, Connes’ quantum Noncommutative Geometry 303, 382 Dimension in see Unbounded Fredholm Module, Dimension of Examples in using singular traces 309 Noncommutative Integral 307, 383, 391, 414, 417 As a hypertrace 393, 408 Extension of the Lebesgue integral see Integral, Of functions Invariant of isospectral deformation 398 Link to classical limits 308, 383, 413 Noncommutative residue Extension of (vector-valued) 304, 306, 337, 352, 365, 385, 396 In Noncommutative Geometry 307, 382, 389 Invariance under isospectral deformation 385, 397 Link to integration see Integral, Of functions Link to singular traces 304, 337, 360, 370 Spectral formula 306, 338, 370, 385 Wodzicki (scalar-valued) 305, 329, 334, 362, 369 Noncommutative Torus 405, 408 Trace on 408 Norm Lp 60 Dual 66 Fatou 59 Operator see Operator norm Quasi 187, 315 Symmetric see Symmetric norm Uniform see Operator norm

450 Operator -measurable 46 Affiliated 46 Bimodule see Von Neumann algebra, Bimodule of Bounded see Bounded operator Cesàro see Cesàro operator Compact see Compact Operator Diagonal 19, 153, 184 Dilation see Dilation operator Expectation see Expectation operator Finite rank 16, 50 Ideal see Ideal Imaginary part 42 Inequalities 422 Localised see Modulated operator, Localised Measurable see Measurable operator Modulated see Modulated operator Normal see Compact operator, Normal or Bounded operator, Normal Nuclear see Operator, Trace class Pseudo-differential see Pseudodifferential operator Real part 42 Space see Symmetric operator space Tauberian see Measurable operator, Tauberian Trace class 29 Operator norm 15, 38, 60 Operator topology Measure 71 Norm 38 Strong 38, 397 Uniform see Operator topology, Norm Weak 38 Weak 212 Peter-Weyl form 398 Projection 39 -finite 43, 64 Abelian 41 Associated to isospectral deformation 397 Finite rank 16, 24 Lattice of 39 Rank one 16, 41

Index

Spectral see Spectral, Projection Support 42 Projection valued measure 41 Pseudo-differential operator 303, 319 Adjoint 319 Bounded extension of 325, 333 Classical 304, 328, 333, 362, 375 Compact extension of 327, 333 Compactly based see Compactly based operator Compactly supported see Compactly supported operator Kernel 320 On a manifold 332 Power of 333 Regularity of 324, 327 Shubin 327, 351 Symbol of see Symbol Trace class 327 Trace on order 0 390 Pseudo-differential operator of order d 303, 329 As a Laplacian modulated operator 351, 364 Eigenvalues 355 Isospectral deformation of 397 Non-uniqueness of traces on see NonMeasurable operator, Pseudodifferential Unique trace on classical see Noncommutative residue, Wodzicki and see Trace Theorem, Connes’ Quantum ergodicity 384, 413 Quasi-nilpotent see Compact operator, Quasi-nilpotent Radon-Nikodym Theorem see Trace, Duality Rapid decrease see Schwartz functions Renormalization (multiplicative) 195 Riemannian geometry 330, 396 Partition of unity 330, 367 Riemannian metric 330, 354, 364, 368, 396 Riesz theorem, on subharmonic functions 171

Index

Schur-Horn problem, Kaftal and Weiss contribution to resolution of 241 Schwartz functions 318, 356 Semifinite Trace see Trace, Semifinite Von Neumann algebra see von Neumann algebra, Semifinite Sequence space lp 24, 61, 185 Lorentz 24, 226 Orlicz 186 Sargent 25, 311 Separable part 164 Weak-lp 24, 185, 187 Singular state On L.H / 384 On l1 or L1 see Extended Limit Singular trace 33, 70, 304, 312, 316, 411 Invariant of isospectral deformation 398 Link to classical limits 308, 385, 411 Link to the Lebesgue integral 305, 338, 372 Link to the noncommutative residue 304, 337, 360, 370, 389 On the weak-l1 ideal 191 Uniqueness of 312 Singular value Function 48, 71, 79, 107 Sequence 19, 23, 49, 77, 153 Singular values (of a compact operator) 19, 23, see also Singular value sequence Product 21 Sum 21, see also Submajorization, Hardy-Littlewood(-Polya) Slow oscillation at infinity see Tauberian, Theorem of Kwee Smooth functions 318, 330, 375 Deformation of see Isospectral deformation, of a smooth function Smooth manifold see Riemannian geometry Sobolev space 324, 333 Connes-Moscovici approach 386

451 Spectral Distribution function 48, 226, 421 Geometry 363 Measure 42, 48 Projection 42 Theorem 19, 41 Spectral triple see Unbounded Fredholm Module Spectrum 17, 233 Sphere (d  1) 328 Square integrable functions see Hilbert space, of square integrable functions State, on a von Neumann algebra 196 ˇ Stone–Cech compactification 195, 214 Subharmonic Function 168 Mapping 168 Submajorization Hardy-Littlewood(-Polya) 58, 69, 83, 312 In the finite-dimensional setting 80 Uniform 79, 88 Symbol Asymptotic expansion of 323 Compactly supported in the first variable 326, 328, 350 Of a Hodge-Laplacian modulated operator 367 Of a Laplacian modulated operator 305, 336, 349, 350, 353, 355, 360 Of a Pseudo-differential operator 320 Principal 321, 329, 333, 353, 367 Symmetric Function space 56, 72, 76, 97, 117 Norm 28, 56, 97 Sequence space 56, 76, 101, 117, 153, 162, 180 Symmetric functional 69, 107, 123 Existence of 108, 120, 123, 142 Hermitian 109 In the Figiel-Kalton sense 152 Lattice of 116, 146 Lifting of from a function space to an operator space see Lifting, Of symmetric functionals from function spaces to operator spaces

452 Link to traces 69, 77, 107, 153 Positive part 109 Singular 70, 108, 123 Symmetric ideal 56 Fully symmetric see Fully symmetric, Ideal Of compact operators 28, 56, 70, 71, 77, 153, 173, 194 Which is not fully or strongly symmetric 72 Symmetric operator space 56, 63, 68, 97, 117, 142 Correspondence with a symmetric function space see Lifting, of symmetric function spaces to symmetric operator spaces Fully symmetric see Fully symmetric, Operator space Lorentz see Lorentz, Operator space Separable part 67, 70, 311 Strongly symmetric 59, 66 Which is an ideal see Symmetric ideal Symmetrically normed ideal see Symmetric ideal, Of compact operators Tangent bundle see Riemannian geometry Tauberian Operator see Measurable operator, Tauberian Theorem of Kwee 384, 392 Variant of Hardy’s theorem, lemma 278 Topology see Operator topology Torus 396, 405 Noncommutative see Noncommutative Torus Trace 26, 68, 316, 340 As a symmetric functional 69, 70, 77, 153 Canonical 29, 44, 185 Construction of 27, 36, 69, 194, 216 Continuous 28, 68, 70, 183, 311 Dixmier see Dixmier trace Duality 62, 67, 73 Existence of 109, 120, 184, 194 Faithful 43 Finite 43

Index

Fully symmetric 69, 194, 312 Hilbert space see Trace, Canonical Hyper see Hypertrace Matrix 22 Normal 43, 61, 68, 70, 217 Normalised 311 On L1,1 190 On a pseudo-differential operator of order d see Noncommutative residue On a pseudo-differential operator of order 0 390 Semifinite 42, 68 Singular see Singular trace Spectrality of see Lidskii formula Wodzicki construction of 195 Trace class operators 28, 29, 62, 185, 311, 384 Pseudo-differential operators as see Pseudo-differential operator, Trace class Trace Theorem 305, 337, 360 Connes’ 304, 362, 372, 380 In Noncommutative Geometry 307, 389 On a manifold 370 Unbounded Fredholm module 382, 386, 391, 394, 413 Dimension of 307, 327, 386, 417 Of a closed Riemannian manifold 396 Of the torus 405 Unitary action (strongly continuous) Of the isometry group 396 Unitary operator see Bounded operator, Unitary Weak-l1 ideal see Ideal, L1,1 Weyl Asymptotic condition (noncommutative geometry) 308, 383, 387, 392, 413 Law 357, 363, 406 Lemmas 21 Relations (noncommutative torus) 408

-function 244

-function residue 222, 244, 263, 276, 314