Invariant ergodic measures and the classification of crossed product $C^\ast$-algebras

Abstract

Let $α: G\curvearrowright X$ be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space $E_G(X)$ is compact and zero-dimensional, we show that the action $α$ has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space $M_G(X)$ of invariant Borel probability measures forms a Bauer simplex. In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is $\mathcal{Z}$-stable. Finally, we discuss some rank properties and provide two classifiability results for crossed products, one of which is based on the work of Elliott and Niu.

17 pages. Major revision. Enhancement of the reader experience by simplifying the proof of Lemma 3.1. Add a discussion of Definition 2.4 and a proof of Proposition 3.8. Update the reference list. To appear in JFA