Internal Measure-Theoretic Phenomena

Abstract

The notion of amenability in its most basic combinatorial sense captures the idea of internal finite approximation from a measure-theoretic perspective. It plays a pivotal role not only in combinatorial and geometric group theory but also in the theory of operator algebras through its various linear manifestations like hyperfiniteness, semidiscreteness, injectivity, and nuclearity. In this section we will review the theory of amenability for discrete groups (see [34, 76] for general references), and then move in Section 14.2 to amenable actions and their reduced crossed products.