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تسجيل مستخدم جديدStable rank of $mathrm{C}(X)rtimesGamma$
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It is shown that, for an arbitrary free and minimal $mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $mathrm{C}(X)rtimesmathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, Gamma)$, where $Gamma$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $mathrm{C}(X)rtimesGamma$ has stable rank one. Moreover, in this case, the C*-algebra $mathrm{C}(X)rtimesGamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.
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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 C
untz comparison of open sets, then $mathrm{mdim}(X, Gamma)=0$ implies that $(mathrm{C}(X) rtimesGamma)otimesmathcal Z cong mathrm{C}(X) rtimesGamma$, 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 $mathrm{C}(X) rtimesGamma$ is classified by the Elliott invariant.
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