1st Edition

Methods of Noncommutative Geometry for Group C*-Algebras

By Do Ngoc Diep Copyright 1999
    368 Pages
    by Chapman & Hall

    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.

    The Scope and an Example
    Multidimensional Orbit Methods
    KK-Theory Invariance IndexC*(G)
    Deformation Quantization and Cyclic Theories
    Bibliographical Remarks
    Classification of MD-Groups
    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
    Multidimensional Quantization
    Induced Representation. Mackey Method of Small Subgroups
    Symplectic Manifolds with Flat Action of Lie Groups
    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


    Do Ngoc Diep