Methods of Noncommutative Geometry for Group C*-Algebras
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Book 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