Quasi-locality for étale groupoids
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914654817615872 |
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| author | Jiang, Baojie Zhang, Jiawen Zhang, Jianguo |
| author_facet | Jiang, Baojie Zhang, Jiawen Zhang, Jianguo |
| contents | Let $\mathcal{G}$ be a locally compact étale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $σ$-compact and amenable, an equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09428 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-locality for étale groupoids Jiang, Baojie Zhang, Jiawen Zhang, Jianguo Operator Algebras Let $\mathcal{G}$ be a locally compact étale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $σ$-compact and amenable, an equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids. |
| title | Quasi-locality for étale groupoids |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2211.09428 |