Fejér property and Galois correspondence for groupoid $C^*$-algebras
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| Format: | Preprint |
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2025
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| _version_ | 1866917089196900352 |
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| author | Anshu Amrutam, Tattwamasi Karmakar, Pradyut |
| author_facet | Anshu Amrutam, Tattwamasi Karmakar, Pradyut |
| contents | We introduce a notion of the Fejér property for topological étale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal étale second countable groupoid satisfying the Fejér property, every closed $C_0(\mathcal{G}^0)$-bimodule $M\subset C_r^*(\mathcal{G})$ is of the form $\overline{C_c(U)}^r$ for some open set $U$. Moreover, we get a Galois correspondence in the sense that every intermediate $C^*$-algebra $\mathcal{B}$ with $C_0(\mathcal{G}^0)\subseteq \mathcal{B}\subseteq C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\leq \mathcal{G}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fejér property and Galois correspondence for groupoid $C^*$-algebras Anshu Amrutam, Tattwamasi Karmakar, Pradyut Operator Algebras Dynamical Systems Functional Analysis General Topology 46L05, 22A22 We introduce a notion of the Fejér property for topological étale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal étale second countable groupoid satisfying the Fejér property, every closed $C_0(\mathcal{G}^0)$-bimodule $M\subset C_r^*(\mathcal{G})$ is of the form $\overline{C_c(U)}^r$ for some open set $U$. Moreover, we get a Galois correspondence in the sense that every intermediate $C^*$-algebra $\mathcal{B}$ with $C_0(\mathcal{G}^0)\subseteq \mathcal{B}\subseteq C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\leq \mathcal{G}$. |
| title | Fejér property and Galois correspondence for groupoid $C^*$-algebras |
| topic | Operator Algebras Dynamical Systems Functional Analysis General Topology 46L05, 22A22 |
| url | https://arxiv.org/abs/2511.14474 |