On Hausdorff covers for non-Hausdorff groupoids
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909557778808832 |
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| author | Brix, Kevin Aguyar Gonzales, Julian Hume, Jeremy B. Li, Xin |
| author_facet | Brix, Kevin Aguyar Gonzales, Julian Hume, Jeremy B. Li, Xin |
| contents | We develop a new approach to non-Hausdorff étale groupoids and their algebras based on Timmermann's construction of Hausdorff covers. As an application, we completely characterise when singular ideals vanish in Steinberg algebras over arbitrary rings. We also completely characterise when $C^*$-algebraic singular ideals have trivial intersection with the non-Hausdorff analogue of subalgebras of continuous, compactly supported functions. This leads to a characterisation when $C^*$-algebraic singular ideals vanish for groupoids satisfying a finiteness condition. Moreover, our approach leads to further sufficient vanishing criteria for singular ideals and reduces questions about simplicity, the ideal intersection property, amenability and nuclearity for non-Hausdorff étale groupoids to the Hausdorff case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23203 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hausdorff covers for non-Hausdorff groupoids Brix, Kevin Aguyar Gonzales, Julian Hume, Jeremy B. Li, Xin Operator Algebras Dynamical Systems Rings and Algebras 22A22, 46L05, 37A55, 16S99 We develop a new approach to non-Hausdorff étale groupoids and their algebras based on Timmermann's construction of Hausdorff covers. As an application, we completely characterise when singular ideals vanish in Steinberg algebras over arbitrary rings. We also completely characterise when $C^*$-algebraic singular ideals have trivial intersection with the non-Hausdorff analogue of subalgebras of continuous, compactly supported functions. This leads to a characterisation when $C^*$-algebraic singular ideals vanish for groupoids satisfying a finiteness condition. Moreover, our approach leads to further sufficient vanishing criteria for singular ideals and reduces questions about simplicity, the ideal intersection property, amenability and nuclearity for non-Hausdorff étale groupoids to the Hausdorff case. |
| title | On Hausdorff covers for non-Hausdorff groupoids |
| topic | Operator Algebras Dynamical Systems Rings and Algebras 22A22, 46L05, 37A55, 16S99 |
| url | https://arxiv.org/abs/2503.23203 |