Morita Equivalence and Continuous-Trace C^*-algebras

by
Iain Raeburn and Dana P. Williams

TABLE OF CONTENTS

• Introduction
• 1 The Algebra of Compact Operators
• 2 Hilbert C^*-Modules
  • 2.1 Hilbert Modules
  • 2.2 Bounded Maps on Hilbert Modules
  • 2.3 Multiplier Algebras
  • 2.4 Induced Representations
• 3 Morita Equivalence
  • 3.1 Imprimitivity Bimodules
  • 3.2 Morita Equivalence
  • 3.3 The Rieffel Correspondence
  • 3.4 The External Tensor Product
• 4 Cohomology
  • 4.1 Sheaf Cohomology
  • 4.2 Fibre Bundles
  • 4.3 The Dixmier-Douady Classification of Locally Trivial Bundles
• 5 Continuous-Trace C^*-Algebras
  • 5.1 C^*-Algebras with Hausdorff Spectrum
  • 5.2 Continuous-Trace C^*-Algebras
  • 5.3 The Dixmier-Douady Classification of Continuous-Trace C^*-Algebras
  • 5.4 Automorphisms of Continuous-Trace C^*-Algebras
  • 5.5 Classification up to Stable Isomorphism
• 6 Applications
  • 6.1 The Brauer Group
  • 6.2 Pull-back C^*-Algebras
  • 6.3 Induced C^*-Algebras
• 7 Epilogue: The Brauer Group and Group Actions
  • 7.1 Dynamical Systems and Crossed Products
  • 7.2 The Equivariant Brauer Group
  • 7.3 The Brauer Group of a Point
  • 7.4 Group Cohomology and Moore Cohomology
  • 7.5 The Brauer Group for Trivial Actions
  • 7.6 The Brauer Group for Free and Proper Actions
  • 7.7 The Structure of the Brauer Group
• A The Spectrum
  • A.1 States and Representations
  • A.2 The Spectrum of a C^*-Algebra
  • A.3 The Dauns-Hofmann Theorem
  • A.4 The State Space of a C^*-Algebra
• B Tensor Products of C^*-Algebras
  • B.1 The Spatial Tensor Product
  • B.2 Fundamental Examples
  • B.3 Other C^*-Norms
  • B.4 C^*-Algebras with Hausdorff Spectrum}{254}
  • B.5 Tensor Products of General C*-Algebras
• C The Imprimitivity Theorem
  • C.1 Haar Measure and Measures on Homogeneous Spaces
  • C.2 Vector-Valued Integration on Groups
  • C.3 The Group C^*-Algebra
  • C.4 The Imprimitivity Theorem
  • C.5 Induced Representations of Groups
  • C.6 The Stone-von Neumann Theorem
• D Miscellany
  • D.1 Direct Limits
  • D.2 The Inductive Limit Topology
• Index
• Bibliography