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

TL;DR

This paper investigates the relationship between invariant ergodic measures and the classification of crossed product C*-algebras, establishing new links under specific measure-theoretic conditions and contributing to the understanding of their structural properties.

Contribution

It demonstrates that under certain conditions, minimal free actions have the small boundary property and that dynamical comparison implies almost finiteness, aiding classification of crossed product C*-algebras.

Findings

Proves the small boundary property under specified measure conditions.

Shows dynamical comparison implies almost finiteness.

Provides new classification results for crossed product C*-algebras.

Abstract

Let $\alpha: G\curvearrowright X$ 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 $\alpha$ 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.