Singular Traces: Theory and Applications

Steven Lord; Fedor Sukochev; Dmitriy Zanin

Published by De Gruyter 2012

Singular Traces

Theory and Applications

Steven Lord , Fedor Sukochev and Dmitriy Zanin

Volume 46 in the series De Gruyter Studies in Mathematics

https://doi.org/10.1515/9783110262551

Overview
Contents

About this book

This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus.

Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace,which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory.

The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces.Singular traces can be equated to symmetric functionals of symmetric
sequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences.

For mathematical readers the text offers fundamental functional analysis results and, due to Nigel Kalton's contribution, a now complete theory of traces on compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and access to the deeper mathematical features of traces on ideals associated to the harmonic sequence. These features, not known and not discussed in general texts on noncommutative geometry, are undoubtably physical and probe to the fascinating heart of classical limits and quantization.

Author / Editor information

Steven Lord, University of Adelaide, Australia; Fedor Sukochev andDmitriy Zanin, University of New South Wales, Sydney, Australia.

Supplementary Materials

Contents

Topics

Frontmatter Publicly Available Download PDF	i
Preface Requires Authentication Unlicensed Licensed Download PDF	v
Notations Requires Authentication Unlicensed Licensed Download PDF	vii
Contents Publicly Available Download PDF	xiii
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Part I. Preliminary Material
Chapter 1. What is a Singular Trace? Requires Authentication Unlicensed Licensed Download PDF	15
Chapter 2. Preliminaries on Symmetric Operator Spaces Requires Authentication Unlicensed Licensed Download PDF	38
Part II. General Theory
Chapter 3. Symmetric Operator Spaces Requires Authentication Unlicensed Licensed Download PDF	75
Chapter 4. Symmetric Functionals Requires Authentication Unlicensed Licensed Download PDF	107
Chapter 5. Commutator Subspace Requires Authentication Unlicensed Licensed Download PDF	153
Chapter 6. Dixmier Traces Requires Authentication Unlicensed Licensed Download PDF	194
Part III. Traces on Lorentz Ideals
Chapter 7. Lidskii Formulas for Dixmier Traces on Lorentz Ideals Requires Authentication Unlicensed Licensed Download PDF	221
Chapter 8. Heat Kernel Formulas and ξ-function Residues Requires Authentication Unlicensed Licensed Download PDF	244
Chapter 9. Measurability in Lorentz Ideals Requires Authentication Unlicensed Licensed Download PDF	272
Part IV. Applications to Noncommutative Geometry
Chapter 10. Preliminaries to the Applications Requires Authentication Unlicensed Licensed Download PDF	303
Chapter 11. Trace Theorems Requires Authentication Unlicensed Licensed Download PDF	336
Chapter 12. Residues and Integrals in Noncommutative Geometry Requires Authentication Unlicensed Licensed Download PDF	382
Appendix A. Operator Results Requires Authentication Unlicensed Licensed Download PDF	420
Bibliography Requires Authentication Unlicensed Licensed Download PDF	429
Index Requires Authentication Unlicensed Licensed Download PDF	445

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate Students, Lecturers, and Researchers in Mathematics and Mathematical Physics; Graduate Students studying Noncommutative Geometry; Specialists in Singular Traces and/or Noncommutative Geometry; Academic Libraries
Pages
- Front matter: 16
- Main content: 452
Keywords: Singular Trace; Dixmier Trace; Noncommutative Residue; Classical Limit; Operator Ideal

eBook

Published: December 19, 2012
ISBN: 9783110262551