Simplicity of twisted C*-algebras of Deaconu--Renault groupoids
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866913345071742976 |
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| author | Armstrong, Becky Brownlowe, Nathan Sims, Aidan |
| author_facet | Armstrong, Becky Brownlowe, Nathan Sims, Aidan |
| contents | We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_02583 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Simplicity of twisted C*-algebras of Deaconu--Renault groupoids Armstrong, Becky Brownlowe, Nathan Sims, Aidan Operator Algebras 46L05 We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions. |
| title | Simplicity of twisted C*-algebras of Deaconu--Renault groupoids |
| topic | Operator Algebras 46L05 |
| url | https://arxiv.org/abs/2109.02583 |