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

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

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