The action of an inverse semigroup on its Stone-Čech compactification
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| Format: | Preprint |
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2025
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| _version_ | 1866915918307655680 |
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| author | Gondek, Joseph P. Z. Starling, Charles |
| author_facet | Gondek, Joseph P. Z. Starling, Charles |
| contents | We initiate the study of the Stone-Čech transformation groupoid $\mathcal{G} = \mathcal{S}\ltimesβ\mathcal{S}$ of an inverse semigroup $\mathcal{S}$. We prove that the properties of being Hausdorff, principal, and effective are all equivalent for $\mathcal{G}$, and give an algebraic condition on $\mathcal{S}$ equivalent to the Hausdorffness of $\mathcal{G}$. We show that the Hausdorffness of Exel's tight groupoid $\mathcal{G}_{\text{tight}}(\mathcal{S})$ is necessary for the Hausdorffness of $\mathcal{G}$. Finally, we clarify the connection between several crossed product constructions involving this groupoid, and show that when it is Hausdorff and $\mathcal{S}$ has the Property (FL) of Lledó and Martínez, then $\mathcal{G}$ is amenable if and only if the reduced C$^*$-algebra $\text{C}^*_r(\mathcal{S})$ is exact. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The action of an inverse semigroup on its Stone-Čech compactification Gondek, Joseph P. Z. Starling, Charles Operator Algebras Category Theory 20M18, 18B40, 46L05 (Primary) 43A65 (Secondary) We initiate the study of the Stone-Čech transformation groupoid $\mathcal{G} = \mathcal{S}\ltimesβ\mathcal{S}$ of an inverse semigroup $\mathcal{S}$. We prove that the properties of being Hausdorff, principal, and effective are all equivalent for $\mathcal{G}$, and give an algebraic condition on $\mathcal{S}$ equivalent to the Hausdorffness of $\mathcal{G}$. We show that the Hausdorffness of Exel's tight groupoid $\mathcal{G}_{\text{tight}}(\mathcal{S})$ is necessary for the Hausdorffness of $\mathcal{G}$. Finally, we clarify the connection between several crossed product constructions involving this groupoid, and show that when it is Hausdorff and $\mathcal{S}$ has the Property (FL) of Lledó and Martínez, then $\mathcal{G}$ is amenable if and only if the reduced C$^*$-algebra $\text{C}^*_r(\mathcal{S})$ is exact. |
| title | The action of an inverse semigroup on its Stone-Čech compactification |
| topic | Operator Algebras Category Theory 20M18, 18B40, 46L05 (Primary) 43A65 (Secondary) |
| url | https://arxiv.org/abs/2508.17516 |