**Memoirs of the American Mathematical Society**

2006;
169 pp;
Softcover

MSC: Primary 46;

Print ISBN: 978-0-8218-3857-0

Product Code: MEMO/180/850

List Price: $71.00

Individual Member Price: $42.60

**Electronic ISBN: 978-1-4704-0454-3
Product Code: MEMO/180/850.E**

List Price: $71.00

Individual Member Price: $42.60

# A Categorical Approach to Imprimitivity Theorems for $C^{*}$-Dynamical Systems

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*Siegfried Echterhoff; S. Kaliszewski; John Quigg; Iain Raeburn*

Imprimitivity theorems provide a fundamental tool for studying
the representation theory and structure of crossed-product
\(C^*\)-algebras. In this work, we show that the Imprimitivity Theorem
for induced algebras, Green's Imprimitivity Theorem for actions of groups, and
Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as
natural equivalences between various crossed-product functors among certain
equivariant categories.

The categories involved have \(C^*\)-algebras with
actions or coactions (or both) of a fixed locally compact group \(G\) as
their objects, and equivariant equivalence classes of right-Hilbert bimodules
as their morphisms. Composition is given by the balanced tensor product of
bimodules.

The functors involved arise from taking crossed products;
restricting, inflating, and decomposing actions and coactions; inducing
actions; and various combinations of these.

Several applications of this categorical approach are also
presented, including some intriguing relationships between the Green and
Mansfield bimodules, and between restriction and induction of
representations.

#### Table of Contents

# Table of Contents

## A Categorical Approach to Imprimitivity Theorems for $C^{*}$-Dynamical Systems

- Contents v6 free
- Introduction 110 free
- Outline 312 free
- Epilogue 615

- Chapter 1. Right-Hilbert Bimodules 918
- Chapter 2. The Categories 3342
- Chapter 3. The Functors 5766
- Chapter 4. The Natural Equivalences 7988
- Chapter 5. Applications 101110
- Appendix A. Crossed Products by Actions and Coactions 117126
- A.1. Tensor products 117126
- A.2. Actions and their crossed products 121130
- A.3. Coactions 126135
- A.4. Slice maps and nondegeneracy 130139
- A.5. Covariant representations and crossed products 132141
- A.6. Dual actions and decomposition coactions 138147
- A.7. Normal coactions and normalizations 139148
- A.8. The duality theorems of Imai-Takai and Katayama 143152
- A.9. Other definitions of coactions 148157

- Appendix B. The Imprimitivity Theorems of Green and Mansfield 151160
- Appendix C. Function Spaces 159168
- Appendix. Bibliography 167176