C∗-algebras and topological dynamics: finite approximation and paradoxicality

Contents Chapter 1. Introduction 5 Chapter 2. Internal measure-theoretic phenomena 13 1. Amenable groups and nuclearity 13 2. Amenable actions, nuclearity, and exactness 16 3. The type semigroup, invariant measures, and pure infiniteness 21 4. The universal minimal system 29 5. Minimal actions, pure infiniteness, and nuclearity 31 Chapter 3. External measure-theoretic phenomena 33 1. Sofic groups, sofic actions, and hyperlinearity 33 2. Entropy 34 3. Combinatorial independence 40 4. Mean dimension 41 Chapter 4. Internal topological phenomena 45 1. Locally finite groups and AF algebras 45 2. Dimension and K-theoretic classification 46 3. Minimal homeomorphisms of zero-dimensional spaces 50 4. Minimal homeomorphisms of finite-dimensional spaces 53 5. Mean dimension and comparison in the Cuntz semigroup 57 Chapter 5. External topological phenomena 63 1. Groups which are locally embeddable into finite groups 63 2. Chain recurrence, residually finite actions, and MF algebras 65 Bibliography 75 3 CHAPTER 1 Introduction The aim of these notes is to survey several recent developments at the crossed product interface of the subjects of C *-algebras and group actions on compact spaces, especially in connection with the classification program for separable nuclear C *-algebras. Groups and group actions have from the beginning provided a rich source of examples in the theory of operator algebras, and the struggle to obtain an algebraic understanding of dynamical phenomena has to a great extent driven, and continues to drive, the development of structure and classification theory for both von Neumann algebras and C *-algebras. While our focus is on the topological realm of C *-algebras, we have nevertheless endeavored to take a broad perspective that incorporates both the measurable and the topological in a unifying framework. This enables us not only to illuminate the conceptual similarities and technical differences between the two sides, but also to emphasize that topological-dynamical and C *-algebraic concepts themselves can range from the more measure-theoretic (like entropy and nuclearity, which involve weak-type approximation of multiplicative structure or norm approximation of linear structure) to the more topological (like periodicity and approximate finite-dimensionality, which involve norm approximation of multiplicative structure). Thinking in such terms can be helpful for predicting and understanding the role of various phenomena in C *-classification theory. One of our major themes is the distinction between internal and external approximation. For a C *-algebra A, internal approximation means modelling the structure of A locally via C *-subalgebras or C *-algebras which map into …