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Lord, Steven / Sukochev, Fedor / Zanin, Dmitriy

Singular Traces

Theory and Applications

Series:De Gruyter Studies in Mathematics 46

    99,95 € / $140.00 / £91.00*

    eBook (PDF)
    Publication Date:
    December 2012
    Copyright year:
    2012
    ISBN
    978-3-11-026255-1
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    Overview

    Aims and Scope

    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.

    Supplementary Information

    Details

    xvi, 452 pages
    Language:
    English
    Type of Publication:
    Monograph
    Keyword(s):
    Singular Trace; Dixmier Trace; Noncommutative Residue; Classical Limit; Operator Ideal

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

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