Topological Dynamics in the Study of C^*-Algebras: Finite Approximations and Dynamic Dimension

Abstract

We investigate properties of dynamical systems motivated by the crossed product construction, specifically finite approximation properties and the dynamic asymptotic dimension. Chapter 1 provides some motivation and historical context with a more in depth overview of each chapter. In chapter 2, we show that equicontinuous actions on Cantor sets are profinite and that equicontinuous actions by finitely generated groups are residually finite. The latter requires some background on representation theory and Lie theory, which we also provide in this chapter. In chapter 3, we show that equicontinuous actions are quasidiagonal and use this to exhibit new examples of group actions whose crossed products have the MF property. Chapter 4 gives some background on the dynamic asymptotic dimen- sion and related concepts in geometric group theory. We show the dynamic asymptotic dimension of actions on profinite completions is closely related to the asymptotic dimension of the acting group’s box spaces. Chapter 5 contains proofs for Hurewicz-type theorems for the dynamic asymptotic dimension, which we use to describe the asymptotic dimension of box spaces of elementary amenable groups. In chapter 6, we give sharp bounds for the dimension of most isometric actions, and use these to completely describe the dimension of translation actions on compact lie groups in terms of the amenability and asymptotic dimension of the acting group.