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

TL;DR

This paper proves that the C*-algebra associated with a minimal homeomorphism of a zero mean dimension space absorbs the Jiang-Su algebra, leading to classification results under unique ergodicity.

Contribution

It establishes Z-stability for the C*-algebra of minimal homeomorphisms with zero mean dimension, extending classification results to this class.

Findings

C*-algebra absorbs Jiang-Su algebra when mean dimension is zero.

C*-algebra tensor square always absorbs Jiang-Su algebra.

Implication for classifiability under unique ergodicity.

Abstract

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. 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.