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

Siegfried Echterhoff Westfälische Wilhelms-Universität, Münster, Germany
S. Kaliszewski Arizona State University, Tempe, AZ
John Quigg Arizona State University, Tempe, AZ
Iain Raeburn University of Newcastle, Newcastle, NSW, Australia
Available Formats:
Electronic ISBN: 978-1-4704-0454-3
Product Code: MEMO/180/850.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view A Categorical Approach to Imprimitivity Theorems for$C^*$-Dynamical Systems Siegfried Echterhoff Westfälische Wilhelms-Universität, Münster, Germany S. Kaliszewski Arizona State University, Tempe, AZ John Quigg Arizona State University, Tempe, AZ Iain Raeburn University of Newcastle, Newcastle, NSW, Australia Available Formats:  Electronic ISBN: 978-1-4704-0454-3 Product Code: MEMO/180/850.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1802006; 169 pp
MSC: Primary 46;

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.

• Chapters
• Introduction
• 1. Right-Hilbert bimodules
• 2. The categories
• 3. The functors
• 4. The natural equivalences
• 5. Applications
• Requests

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Volume: 1802006; 169 pp
MSC: Primary 46;

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.

• Chapters
• Introduction
• 1. Right-Hilbert bimodules
• 2. The categories
• 3. The functors
• 4. The natural equivalences
• 5. Applications
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
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