Methods of Noncommutative Geometry for Group C*-Algebras,9781584880196

Methods of Noncommutative Geometry for Group C*-Algebras

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ISBN13: 9781584880196
ISBN10: 1584880198
Format: Nonspecific Binding
Pub. Date: 1999-12-06
Publisher(s): Chapman & Hall/
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Summary

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

Preface
Introduction
1(26)
The Scope and an Example
1(8)
The Problem
1(4)
BDF K-Homology Functor
5(2)
Topological Invariant Index
7(2)
Multidimensional Orbit Methods
9(8)
Multidimensional Quantization
9(7)
Category O and Globalization of Harish-Chandra Modules
16(1)
KK-theory Invariant Index C*(G)
17(9)
About KK-Functors
17(3)
Construction and Reduction of the K-theory Invariant Index C*(G)
20(6)
Deformation Quantization and Cyclic Theories
26(1)
Star-products and Star-representations
26(1)
Periodic Cyclic Homology
26(1)
Chern Characters
26(1)
Bibliographical Remarks
26(1)
I Elementary Theory: An Overview Based on Examples 27(118)
Classification of MD-Groups
39(10)
Definitions
39(2)
MD-criteria
41(1)
Classification Theorem
42(5)
Bibliographical Remarks
47(2)
The Structure of C*-algebras of MD-groups
49(34)
The C*-algebra of Aff R
49(19)
Statement of Theorems
49(2)
Proof of Theorem 3.1
51(1)
Proof of Theorem 3.2
52(4)
Proof of Theorem 3.3
56(12)
The Structure of C* (Aff C)
68(14)
Bibliographical Remarks
82(1)
Classification of MD4-groups
83(40)
Real Diamond Group and Semidirect Products R x H3
83(2)
Classification Theorem
85(8)
Description of the Coadjoint Orbits
93(19)
Some Remarks about the Coadjoint Representation
93(4)
Description of Coadjoint Orbits
97(15)
Measurable MD4-foliations
112(10)
Measurable Foliations after A. Connes
112(2)
Measurable MD4-foliations
114(4)
Topological Classification of MD4-foliations
118(4)
Bibliographical Remarks
122(1)
The Structure of C*-algebras of MD4-foliations
123(22)
C*-algebras of Measurable Foliations
123(5)
Holonomy Group of Foliations
123(1)
Half-density Bundle
124(1)
C*-algebras of Measurable Foliations
125(3)
Connes-Thom Isomorphism
128(1)
The C*-algebras of Measurable MD4-foliations
128(15)
C*-algebras of MD4-foliations and Bundle Type
129(1)
C*-algebras of MD4-foliations of Crossed Product Type
129(14)
Bibliographical Remarks
143(2)
II Advanced Theory: Multidimensional Quantization and the Index of Group C*-algebras 145(196)
Multidimensional Quantization
147(38)
Induced Representations. Mackey Method of Small Subgroups
147(9)
Criterion of Inductibility
147(6)
The Mackey Method of Small Subgroups
153(1)
Projective Representations and Mackey Obstructions
154(2)
Symplectic Manifolds with Flat Action of Lie Groups
156(9)
Flat Action
157(6)
Classification
163(2)
Prequantization
165(7)
Quantization Procedure
165(6)
Application
171(1)
Polarization
172(11)
Some Ideas from Physics
172(3)
(F,σ)-polarizations and Polarizations
175(3)
Complex Polarizations
178(2)
Weak Lagrangian Distributions
180(1)
Duflo Data
180(3)
Bibliographical Remarks
183(2)
Partially Invariant Holomorphically Induced Representations
185(40)
Holomorphically Induced Representations. Lie Derivative
185(8)
Partially Invariant Holomorphically Induced Representations
185(4)
Unitarizations
189(1)
Lie Derivation
190(3)
Irreducible Representations of Nilpotent Lie Groups
193(7)
Duflo Construction
194(2)
Metaplectic Shale-Weil Representation
196(2)
Irreducible Unitary Representations of Nilpotent Lie Groups
198(1)
Irreducible Representations of Extensions of Nilpotent Lie Groups
199(1)
Representations of Connected Reductive Groups
200(12)
Harish-Chandra Construction of π(Λ,λ)
204(2)
(Possibly Nonconnected) Reductive Groups
206(3)
The Induction Procedure for General (Separable) Lie Groups
209(3)
Representations of Almost Algebraic Lie Groups
212(6)
Coisotropic Subalgebras
213(2)
Irreducible Representations
215(3)
The Trace Formula and the Plancherel Formula
218(5)
Trace Formula
218(2)
Plancherel Formula for Unimodular Groups
220(3)
Bibliographical Remarks
223(2)
Reduction, Modification, and Supervision
225(84)
Reduction to the Semisimple or Reductive Cases
225(10)
Coisotropic Tangent Distributions
225(3)
(σ,XF)-polarizations
228(3)
Induced Representations Obtained from the Solvable or Unipotent Polarizations
231(3)
Unitary Representations Arising in the Reduction of the Multidimensional Quantization Procedure
234(1)
Multidimensional Quantization and U(1)-covering
235(23)
Positive Polarizations
235(4)
Lifted Characters
239(4)
Induced Representations
243(4)
Multidimensional Quantization
247(2)
U(1)-covering of Radicals and Semisimple or Reductive Data
249(5)
Induction from Semisimple Data
254(2)
A Reduction of the Multidimensional Quantization Procedure on the U(1)-covering
256(2)
Globalization over U(1)-coverings
258(15)
Classical Constructions and Three Geometric Complexes
258(4)
Isomorphisms of Cohomologies
262(3)
Maximal Real Polarizations and Change of Polarizations
265(8)
Quantization of Mechanical Systems with Supersymmetry
273(34)
Hilbert Superbundles with Connection
273(16)
Quantization Superoperators
289(13)
Superpolarizations and Induced Representations
302(5)
Bibliographical Remarks
307(2)
Index of Type I C*-algebras
309(22)
Compact Type Ideals in Type I C*-algebras
309(3)
Canonical Composition Series
312(3)
Index of Type I C*-algebras
315(4)
Compactness Criteria for Group Cast;-algebras
319(7)
Compactness Criteria
320(6)
Application to Lie Group Representations
326(3)
The Case of Solvable Lie Groups
326(1)
Generic Representations of Reductive Lie Groups
327(2)
Bibliographical Remarks
329(2)
Invariant Index of Group C*-algebras
331(10)
The Structure of Group C*-algebras
331(3)
Construction of Index C*(G)
334(3)
Reduction of the Indices
337(2)
General Remarks on Computation of Indices
339(1)
Bibliographical Remarks
339(2)
References 341

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