**Memoirs of the American Mathematical Society**

1993;
109 pp;
Softcover

MSC: Primary 46;
Secondary 22

Print ISBN: 978-0-8218-2545-7

Product Code: MEMO/101/484

List Price: $31.00

Individual Member Price: $18.60

**Electronic ISBN: 978-1-4704-0061-3
Product Code: MEMO/101/484.E**

List Price: $31.00

Individual Member Price: $18.60

# Duality for Actions and Coactions of Measured Groupoids on von Neumann Algebras

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*Takehiko Yamanouchi*

Through classification of compact abelian group actions on semifinite injective factors, Jones and Takesaki introduced the notion of an action of a measured groupoid on a von Neumann algebra, which has proven to be an important tool for this kind of analysis. Elaborating on this notion, this work introduces a new concept of a measured groupoid action that may fit more perfectly into the groupoid setting. Yamanouchi also shows the existence of a canonical coproduct on every groupoid von Neumann algebra, which leads to a concept of a coaction of a measured groupoid. Yamanouchi then proves duality between these objects, extending Nakagami-Takesaki duality for (co)actions of locally compact groups on von Neumann algebras.

#### Table of Contents

# Table of Contents

## Duality for Actions and Coactions of Measured Groupoids on von Neumann Algebras

- Table of Contents v6 free
- Abstract vi7 free
- § 0. Introduction 18 free
- § 1. Relative tensor products of Hilbert spaces over abelian von Neumann algebras 714 free
- § 2. Coproducts of groupoid von Neumann algebras 1421
- § 3. Actions and coactions of measured groupoids on von Neumann algebras 2936
- § 4. Crossed products by groupoid actions and their dual coactions 3340
- § 5. Crossed products by groupoid coactions and their dual actions 5663
- § 6. Duality for actions on von Neumann algebras 6875
- § 7. Duality for integrable coactions on von Neumann algebras 8087
- § 8. Examples of actions and coactions of measured groupoids on von Neumann algebras 9097
- References 106113

#### Readership

Research mathematicians.