Methods of Noncommutative Geometry for Group C*-Algebras: 1st Edition (Paperback) book cover

Methods of Noncommutative Geometry for Group C*-Algebras

1st Edition

By Do Ngoc Diep

Chapman and Hall/CRC

368 pages

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Paperback: 9781584880196
pub: 1999-12-06
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Description

The description of the structure of group C*-algebras is a difficult problem, but relevant to important new developments in mathematics, such as non-commutative geometry and quantum groups. Although a significant number of new methods and results have been obtained, until now they have not been available in book form.

This volume provides an introduction to and presents research on the study of group C*-algebras, suitable for all levels of readers - from graduate students to professional researchers. The introduction provides the essential features of the methods used. In Part I, the author offers an elementary overview - using concrete examples-of using K-homology, BFD functors, and KK-functors to describe group C*-algebras. In Part II, he uses advanced ideas and methods from representation theory, differential geometry, and KK-theory, to explain two primary tools used to study group C*-algebras: multidimensional quantization and construction of the index of group C*-algebras through orbit methods.

The structure of group C*-algebras is an important issue both from a theoretical viewpoint and in its applications in physics and mathematics. Armed with the background, tools, and research provided in Methods of Noncommutative Geometry for Group C*-Algebras, readers can continue this work and make significant contributions to perfecting the theory and solving this problem.

Table of Contents

Introduction

The Scope and an Example

Multidimensional Orbit Methods

KK-Theory Invariance IndexC*(G)

Deformation Quantization and Cyclic Theories

Bibliographical Remarks

ELEMENTARY THEORY: AN OVERVIEW BASED ON EXAMPLES

Classification of MD-Groups

Definitions

MD Criteria

Classification Theorem

Bibliographical Remarks

The Structure of C*-Algebras of MD-Groups

The C*-Algebra of Aff R

The Structure of C*(Aff C)

Bibliographical Remarks

Classification of MD4-Groups

Real Diamond Group and Semi-Direct Products R x H3

Classification Theorem

Description of the Co-Adjoint Orbits

Measurable MD4-Foliation

Bibliographical Remarks

The Structure of C*-Algebras of MD4-Foliations

C*-Algebras of Measurable Foliations

The C*-Algebras of Measurable MD4-Foliations

Bibliographic Remarks

ADVANCED THEORY: MULTIDIMENSIONAL QUANTIZATION AND INDEX OF GROUP C*-ALGEBRAS

Multidimensional Quantization

Induced Representation. Mackey Method of Small Subgroups

Symplectic Manifolds with Flat Action of Lie Groups

Prequantization

Polarization

Bibliographical Remarks

Partially Invariant Holomorphly Induced Representations

Holomorphly Induced Representations. Lie Derivative

The Irreducible Representations of Nilpotent Lie Groups

Representations of Connected Reductive Groups

Representations of Almost Algebraic Lie Groups

The Trace Formula and the Plancher'el Formula

Bibliographical Remarks

Reduction, Modification, and Superversion

Reduction to the Semi-Simple or Reductive Cases

Multidimensional Quantization and U(1)-Covering

Globalization over U(1)-Coverings

Quantization of Mechanical Systems with Supersymmetry

Bibliographical Remarks

Index of Type I C*-Algebras

Compact Type Ideals in Type I C*-Algebras

Canonical Composition series

Index of Type I C*-Algebras

Application to Lie Group Representations

Bibliographical Remarks

Invariant Index of Group C*-Algebras

The Structure of Group C*-Algebras

Construction of IndexC*(G)

Reduction of the Indices

General Remarks on Computation of Indices

Bibliographical Remarks

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis