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 (URP) and Cuntz comparison of open sets (COS), 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 of $(X, \Gamma )$, $\mathcal Z$ is the Jiang-Su algebra, and $\mathrm {C}(X) \rtimes \Gamma$ is the transformation group C*-algebra of $(X, \Gamma )$. 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.

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中文翻译：

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𝒵-变换群C*-代数的稳定性

摘要：设$(X, \Gamma )$ 是一个自由极小拓扑动力系统，其中$X$ 是一个可分的紧致Hausdorff 空间，$\Gamma$ 是一个可数的无限离散服从群。证明如果 $(X, \Gamma )$ 具有统一 Rokhlin 性质（URP）和开集 Cuntz 比较（COS），则 $\mathrm {mdim}(X, \Gamma )=0$ 意味着 $ (\mathrm {C}(X) \rtimes \Gamma )\otimes \mathcal Z \cong \mathrm {C}(X) \rtimes \Gamma$，其中 $\mathrm {mdim}$ 是 $( X, \Gamma )$, $\mathcal Z$ 是江苏代数，$\mathrm {C}(X) \rtimes \Gamma$ 是$(X, \Gamma ) 的变换群C*-代数$. 特别地，在这种情况下， $\mathrm {mdim}(X, \Gamma )=0$ 意味着 C*-代数 $\mathrm {C}(X) \rtimes \Gamma$ 是由 Elliott 不变量分类的。

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