Essential groupoid amenability and nuclearity of groupoid C*-algebras
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908324155359232 |
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| author | Buss, Alcides Martínez, Diego |
| author_facet | Buss, Alcides Martínez, Diego |
| contents | We give an alternative construction of the essential $C^*$-algebra of an étale groupoid, along with an ``amenability'' notion for such groupoids that is implied by the nuclearity of this essential $C^*$-algebra. In order to do this we first introduce a maximal version of the essential $C^*$-algebra, and prove that every function with dense co-support can only be supported on the set of ``dangerous'' arrows. We then introduce an essential amenability condition for a groupoid, which is (strictly) weaker than its (topological) amenability. As an application, we describe the Bruce-Li algebras arising from algebraic actions of cancellative semigroups as exotic essential $C^*$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Essential groupoid amenability and nuclearity of groupoid C*-algebras Buss, Alcides Martínez, Diego Operator Algebras 20L05, 46L52, 46L05 We give an alternative construction of the essential $C^*$-algebra of an étale groupoid, along with an ``amenability'' notion for such groupoids that is implied by the nuclearity of this essential $C^*$-algebra. In order to do this we first introduce a maximal version of the essential $C^*$-algebra, and prove that every function with dense co-support can only be supported on the set of ``dangerous'' arrows. We then introduce an essential amenability condition for a groupoid, which is (strictly) weaker than its (topological) amenability. As an application, we describe the Bruce-Li algebras arising from algebraic actions of cancellative semigroups as exotic essential $C^*$-algebras. |
| title | Essential groupoid amenability and nuclearity of groupoid C*-algebras |
| topic | Operator Algebras 20L05, 46L52, 46L05 |
| url | https://arxiv.org/abs/2501.01775 |