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arXiv 2211.00547math.OA
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Alexandrov 群胚与扭曲群胚 $\mathrm{C}^*$-代数的核维数

Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

Kristin Courtney, Anna Duwenig, Magdalena C. Georgescu, Astrid an Huef, Maria Grazia Viola

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41 pages; minor changes compared to v3; to appear in J. Funct. Anal
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AI总结：

本文考虑étale群胚上的扭曲，证明了主群胚约化扭曲群胚 $\mathrm{C}^*$-代数的核维数界限，推广了已有定理，并应用于具有闭轨道和连续变化 Abel 稳定子群的群胚 $\mathrm{C}^*$-代数。

摘要：

我们考虑étale群胚 $G$ 上的扭曲 $E$。当 $G$ 是主群胚时，我们证明约化扭曲群胚 $\mathrm{C}^*$-代数的核维数受限于一个依赖于 $G$ 的动态渐近维数及其单位空间的拓扑覆盖维数的数值。这推广了 Guentner、Willett 和 Yu 关于 $G$ 的 $\mathrm{C}^*$-代数的一个类似定理。我们的证明通过构造 $G$ 和 $E$ 的“群胚单位化” $\widetilde{G}$ 和 $\widetilde{E}$（使得 $\widetilde{E}$ 是 $\widetilde{G}$ 上的扭曲），归约到 $G$ 具有紧单位空间的幺正情形。$\widetilde{G}$ 的构造适用于不一定是主群胚的 r-离散（因此是étale）群胚 $G$。当 $G$ 是étale群胚时，$G$ 和 $\widetilde{G}$ 的动态渐近维数一致。我们证明 $G$ 上扭曲的全扭曲群胚 $\mathrm{C}^*$-代数和约化扭曲群胚 $\mathrm{C}^*$-代数的最小单位化同构于 $\widetilde{G}$ 上扭曲的扭曲群胚 $\mathrm{C}^*$-代数。我们应用关于扭曲群胚 $\mathrm{C}^*$-代数核维数的结果，获得了具有闭轨道且稳定子群连续变化的 Abel 群的étale群胚 $\mathrm{C}^*$-代数核维数的类似界限。

英文摘要：

We consider a twist $E$ over an étale groupoid $G$. When $G$ is principal, we prove that the nuclear dimension of the reduced twisted groupoid $\mathrm{C}^*$-algebra is bounded by a number depending on the dynamic asymptotic dimension of $G$ and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the $\mathrm{C}^*$-algebra of $G$. Our proof uses a reduction to the unital case where $G$ has compact unit space, via a construction of ``groupoid unitizations'' $\widetilde{G}$ and $\widetilde{E}$ of $G$ and $E$ such that $\widetilde{E}$ is a twist over $\widetilde{G}$. The construction of $\widetilde G$ is for r-discrete (hence étale) groupoids $G$ which are not necessarily principal. When $G$ is étale, the dynamic asymptotic dimension of $G$ and $\widetilde{G}$ coincide. We show that the minimal unitizations of the full and reduced twisted groupoid $\mathrm{C}^*$-algebras of the twist over $G$ are isomorphic to the twisted groupoid $\mathrm{C}^*$-algebras of the twist over $\widetilde{G}$. We apply our result about the nuclear dimension of the twisted groupoid $\mathrm{C}^*$-algebra to obtain a similar bound on the nuclear dimension of the $\mathrm{C}^*$-algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.

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