Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2022
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| _version_ | 1866913236624867328 |
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| author | Courtney, Kristin Duwenig, Anna Georgescu, Magdalena C. Huef, Astrid an Viola, Maria Grazia |
| author_facet | Courtney, Kristin Duwenig, Anna Georgescu, Magdalena C. Huef, Astrid an Viola, Maria Grazia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00547 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras Courtney, Kristin Duwenig, Anna Georgescu, Magdalena C. Huef, Astrid an Viola, Maria Grazia Operator Algebras 46L05, 22A22 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. |
| title | Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras |
| topic | Operator Algebras 46L05, 22A22 |
| url | https://arxiv.org/abs/2211.00547 |