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arXiv · 1711.00886

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

Xin Ma
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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.

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Xin Ma. 2018-06-28. Invariant ergodic measures and the classification of crossed product $C^\ast$-algebras. https://arxiv.org/abs/1711.00886

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