$\mathcal Z$-stability of $\mathrm{C}(X)\rtimes\Gamma$

TL;DR

This paper proves that under certain conditions, the crossed product C*-algebra arising from a minimal dynamical system with zero mean dimension absorbs the Jiang-Su algebra, leading to its classification by Elliott invariants.

Contribution

It establishes $ ext{Z}$-stability for crossed product C*-algebras of minimal dynamical systems with zero mean dimension, given the Uniform Rokhlin Property and Cuntz comparison.

Findings

If $(X, olinebreak \Gamma)$ has the Uniform Rokhlin Property and Cuntz comparison, then zero mean dimension implies $ ext{Z}$-stability.

$ ext{Z}$-stability leads to classification of the C*-algebra by Elliott invariants.

The result applies to free, minimal actions of amenable groups on compact spaces.

Abstract

Let $(X, \Gamma)$ be a free and minimal topological dynamical system, where $X$ is a separable compact Hausdorff space and $\Gamma$ is a countable infinite discrete amenable group. It is shown that if $(X, \Gamma)$ has the Uniform Rokhlin Property and Cuntz comparison of open sets, then $\mathrm{mdim}(X, \Gamma)=0$ implies that $(\mathrm{C}(X) \rtimes\Gamma)\otimes\mathcal Z \cong \mathrm{C}(X) \rtimes\Gamma$, where $\mathrm{mdim}$ is the mean dimension and $\mathcal Z$ is the Jiang-Su algebra. In particular, in this case, $\mathrm{mdim}(X, \Gamma)=0$ implies that the C*-algebra $\mathrm{C}(X) \rtimes\Gamma$ is classified by the Elliott invariant.