Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension

1. Front Matter

   Pages i-x

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2. A Primer on Crossed Products

   1. Front Matter

      Pages 1-4

      PDF

   2. What is a Crossed Product?

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 5-16

   3. Induced Representations and Imprimitivity Theorems

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 17-27

   4. Ideal Structure

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 29-36

   5. Type Theory, Actions on the Compacts and the Effros–Hahn Theory

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 37-44

   6. Tensor Products, K-Theory and Rotation Algebras

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 45-52

   7. The Roads Not Taken

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 53-57

3. Hausdor Étale Groupoids and Their C*-algebras

   1. Front Matter

      Pages 59-61

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   2. Introduction

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 63-66

   3. Etale Groupoids

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 67-76

   4. C*-algebras and Equivalence

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 77-88

   5. Fundamental Structure Theory

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 89-102

   6. Cartan Pairs, and Dixmier–Douady Theory for Fell Algebras

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 103-120

4. Introduction to Rokhlin Dimension

   1. Front Matter

      Pages 121-123

      PDF

   2. Introduction

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 125-127

   3. Classical Dynamical Systems

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 129-135

   4. Classification Theory and Nuclear Dimension

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 137-140

   5. Rokhlin Dimension for Finite Groups

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 141-145

   6. Single Automorphisms

      • Aidan Sims, Gábor Szabó, Dana Williams

      Pages 147-153

This book collects the notes of the lectures given at the Advanced Course on Crossed Products, Groupoids, and Rokhlin dimension, that took place at the Centre de Recerca Matemàtica (CRM) from March 13 to March 17, 2017. The notes consist of three series of lectures. The first one was given by Dana Williams (Dartmouth College), and served as an introduction to crossed products of C*-algebras and the study of their structure. The second series of lectures was delivered by Aidan Sims (Wollongong), who gave an overview of the theory of topological groupoids (as a model for groups and group actions) and groupoid C*-algebras, with particular emphasis on the case of étale groupoids. Finally, the last series was delivered by Gábor Szabó (Copenhagen), and consisted of an introduction to Rokhlin type properties (mostly centered around the work of Hirshberg, Winter and Zacharias) with hints to the more advanced theory related to groupoids.

• crossed products
• topological groupoid, groupoid C*-algebra
• amenability
• Rokhlin dimension
• nuclear dimension

• Universitat Autònoma de Barcelona, Departament de Matemàtiques, Bellaterra, Spain

  Francesc Perera

• School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia

  Aidan Sims

• Department of Mathematics, KU Leuven, Leuven, Belgium

  Gábor Szabó

• Department of Mathematics, Dartmouth College, Hanover, USA

  Dana Williams

Aidan Sims is a professor at the University of Wollongong in Australia.

Gábor Szabó is a research professor at KU Leuven in Belgium.

Dana P. Williams is a professor of mathematics at the Dartmouth College in Hanover, USA.

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