The C∗-algebra of a minimal homeomorphism of zero mean dimension

Abstract

Let $X$ be an infinite metrizable compact space, and let $\sigma :X\to X$ be a minimal homeomorphism. Suppose that $(X,\sigma )$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}\left(X\right){⋊}_{\sigma }\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes \mathcal{Z}$. This implies that $A$ is classifiable when $(X,\sigma )$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.