Invariant measures and traces on groupoid $\mathrm{C}^\ast$-algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915833914064896 |
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| author | Miller, Alistair Scarparo, Eduardo |
| author_facet | Miller, Alistair Scarparo, Eduardo |
| contents | We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $μ$.
We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$.
We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids.
As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04020 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Invariant measures and traces on groupoid $\mathrm{C}^\ast$-algebras Miller, Alistair Scarparo, Eduardo Operator Algebras 46L05 We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $μ$. We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state. |
| title | Invariant measures and traces on groupoid $\mathrm{C}^\ast$-algebras |
| topic | Operator Algebras 46L05 |
| url | https://arxiv.org/abs/2603.04020 |