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Induced Representations of Locally Compact Groups

Book Description

The dual space of a locally compact group G consists of the equivalence classes of irreducible unitary representations of G. This book provides a comprehensive guide to the theory of induced representations and explains its use in describing the dual spaces for important classes of groups. It introduces various induction constructions and proves the core theorems on induced representations, including the fundamental imprimitivity theorem of Mackey and Blattner. An extensive introduction to Mackey analysis is applied to compute dual spaces for a wide variety of examples. Fell's contributions to understanding the natural topology on the dual are also presented. In the final two chapters, the theory is applied in a variety of settings including topological Frobenius properties and continuous wavelet transforms. This book will be useful to graduate students seeking to enter the area as well as experts who need the theory of unitary group representations in their research.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. 1 - Basics
    1. 1.1 Locally Compact Groups
    2. 1.2 Examples
    3. 1.3 Coset Spaces and Quasi-Invariant Measures
    4. 1.4 Representations
    5. 1.5 Representations of L1(G) and Functions of Positive Type
    6. 1.6 C*-Algebras and Weak Containment of Representations
    7. 1.7 Abelian Locally Compact Groups
    8. 1.8 Notes and References
  7. 2 - Induced Representations
    1. 2.1 Inducing from an Open Subgroup
    2. 2.2 Conditions for Irreducibility of Induced Representations
    3. 2.3 The Induced Representation in General
    4. 2.4 Other Realizations and Positive Definite Measures
    5. 2.5 The Affine Group and SL(2, ℝ)
    6. 2.6 Some Basic Properties of Induced Representations
    7. 2.7 Induction in Stages
    8. 2.8 Tensor Products of Induced Representations
    9. 2.9 Frobenius Reciprocity
    10. 2.10 Notes and References
  8. 3 - The Imprimitivity Theorem
    1. 3.1 Systems of Imprimitivity
    2. 3.2 Induced Systems of Imprimitivity
    3. 3.3 The Imprimitivity Theorem
    4. 3.4 Proof of the Imprimitivity Theorem: The General Case
    5. 3.5 Notes and References
  9. 4 - Mackey Analysis
    1. 4.1 Mackey Analysis for Almost Abelian Groups
    2. 4.2 Orbits in the Dual of an Abelian Normal Subgroup
    3. 4.3 Mackey Analysis for Abelian Normal Subgroups
    4. 4.4 Examples: Some Solvable Groups
    5. 4.5 Examples: Action by Compact Groups
    6. 4.6 Limitations on Mackey’s Theory
    7. 4.7 Cocycles and Cocycle Representations
    8. 4.8 Mackey’s Theory for a Nonabelian Normal Subgroup
    9. 4.9 Notes and References
  10. 5 - Topologies on Dual Spaces
    1. 5.1 The Inner Hull–Kernel Topology
    2. 5.2 The Subgroup C*-Algebra
    3. 5.3 Subgroup Representation Topology and Functions of Positive Type
    4. 5.4 Continuity of Inducing and Restricting Representations
    5. 5.5 Examples: Nilpotent and Solvable Groups
    6. 5.6 The Topology on the Dual of a Motion Group
    7. 5.7 Examples: Motion Groups
    8. 5.8 The Primitive Ideal Space of a Two-Step Nilpotent Group
    9. 5.9 Notes and References
  11. 6 - Topological Frobenius Properties
    1. 6.1 Amenability and Induced Representations
    2. 6.2 Basic Definitions and Inheritance Properties
    3. 6.3 Motion Groups
    4. 6.4 Property (FP) for Discrete Groups
    5. 6.5 Nilpotent Groups
    6. 6.6 Notes and References
  12. 7 - Further Applications
    1. 7.1 Asymptotic Properties of Irreducible Representations of Motion Groups
    2. 7.2 Projections in L1(G)
    3. 7.3 Generalizations of the Wavelet Transform
    4. 7.4 Notes and References
  13. Bibliography
  14. Index