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About this book

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.

Table of Contents

Frontmatter

A Primer on Crossed Products

Frontmatter

Chapter 1. What is a Crossed Product?

Abstract
Dynamical systems have their origin in statistical mechanics. (Let G=R and think of α as the time evolution of the system.) Here we will simply agree that interest in dynamical systems and their associated C*-algebras is a prerequisite for this course.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 2. Induced Representations and Imprimitivity Theorems

Abstract
Some knowledge of Hilbert modules and their adjointable operators is a prerequisite for this lecture series. Chapter two of [RW98] and/or a familiarity with Lance's book [Lan95] is plenty of background.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 3. Ideal Structure

Abstract
The theory of amenable groups is classical and vast. A very short summary can be found in [Wil07, Appendix A].
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 4. Type Theory, Actions on the Compacts and the Effros–Hahn Theory

Abstract
There is a very long list of conditions on a C*-algebra A that are equivalent to A being GCR { see [Ped79, Theorem 6.8.7] for one such. This is related to the Mackey{Glimm dichotomy we talked about earlier.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 5. Tensor Products, K-Theory and Rotation Algebras

Abstract
A primer on C*-tensor products can be found in [RW98, Appendix B].
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 6. The Roads Not Taken

Abstract
Since these notes were meant only as a gentle introduction to a vast subject, I have not tried to push the boundary all the way to current research problems. I have not even mentioned the numerous generalizations actively studied in the literature.
Aidan Sims, Gábor Szabó, Dana Williams

Hausdor Étale Groupoids and Their C*-algebras

Frontmatter

Chapter 7. Introduction

Abstract
Groupoids are algebraic objects that behave like a group except that the multiplication operation is only partially defined. Topological groupoids provide a useful unifying model for groups and group actions, and equivalence relations induced by continuous maps between topological spaces. They also provide a good algebraic model for the quotient of a topological space by a group or semigroup action in instances where the quotient space itself is, topologically, poorly behaved { for example, the quotient of a shift-space determined by the shift map, or the quotient of the circle by an irrational rotation.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 8. Etale Groupoids

Abstract
The following definition of a groupoid comes from [24] (see [31, page 7]); Hahn himself attributes it to a conversation with G. Mackey. This is a fairly minimal set of axioms, so optimal for the purposes of checking whether a given object is a groupoid,
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 9. C*-algebras and Equivalence

Abstract
In this section, we will associate two C*-algebras to each éetale groupoid. As with groups and dynamical systems, each groupoid has both a reduced C*-algebra and a full C*-algebra.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 10. Fundamental Structure Theory

Abstract
The theory of amenability for groupoids is complicated; it could easily be a fivehour course all by itself. So we are going to skate over the top of it here. Most of what appears here is taken from [2].
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 11. Cartan Pairs, and Dixmier–Douady Theory for Fell Algebras

Abstract
In this chapter we first discuss the beautiful reconstruction theorem of Renault [39] that shows that an effective groupoid and twist can be recovered from the associated twisted groupoid algebra.
Aidan Sims, Gábor Szabó, Dana Williams

Introduction to Rokhlin Dimension

Frontmatter

Chapter 12. Introduction

Abstract
The classical Rokhlin lemma, sometimes also called Kakutani{Rokhlin lemma, is a fundamental result in ergodic theory. It states that an aperiodic measurepreserving dynamical system can be decomposed into an arbitrarily high tower of measurable sets and a remainder of arbitrarily small measure.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 13. Classical Dynamical Systems

Abstract
In this part, we will treat the classical Rokhlin lemma. For its generalizations and importance within ergodic theory, the interested reader may consult the excellent recent book of Kerr{Li [27].
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 14. Classification Theory and Nuclear Dimension

Abstract
Let us now switch to our main field of interest, namely C*-algebra theory. Although it would have been easy to fll the rest of these notes with applications of the Rokhlin property to classification of automorphisms on C*-algebras, we will instead focus on its applications to the structure of crossed products, and the key ideas giving rise to the notion of Rokhlin dimension.
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 15. Rokhlin Dimension for Finite Groups

Abstract
In order to capture the main idea for the relevance of Rokhlin-type properties to nuclear dimension, it is useful to step back from the single automorphism case and focus first on finite groups. (Note that everything in this section can be generalized to compact groups [20, 14, 13].)
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 16. Single Automorphisms

Abstract
The following version of Rokhlin dimension for single automorphisms is the naive generalization of Rokhlin dimension for finite group actions we have just seen. Note that there is some freedom of choice how to set up this definition, and there are several versions treated in the original paper of Hirshberg{Winter{Zacharias [21].
Aidan Sims, Gábor Szabó, Dana Williams

Chapter 17. Outlook: Generalizations

Abstract
In analogy to how we defined Rokhlin dimension for Z using its finite quotients, it is possible to define Rokhlin dimension for residually finite groups [42, 44].
Aidan Sims, Gábor Szabó, Dana Williams

Backmatter

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