A continuous field of Roe algebras
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913718798909440 |
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| author | Manuilov, V. |
| author_facet | Manuilov, V. |
| contents | Let $X$ be a metric measure space. A Delone subset $D\subset X$ is a uniformly discrete set coarsely equivalent to $X$. We consider the space $\mathcal D_F$ of controlled Delone subsets of $X$ with an appropriate metric, and show that it, together with $X$ itself, is a compact space. By assigning to each point $D$ of $\mathcal D_F$ (resp., to $X$) the uniform Roe algebra $C^*_u(D)$ (resp., the \u Spakula's version $C_k^*(X)$ of the Roe algebra of $X$) we get a tautological family of $C^*$-algebras. For a sequence $\{D_n\}_{n\in\mathbb N}$ of controlled Delone subsets convergent to $X$ we show that the corresponding uniform Roe algebras $C^*_u(D_n)$, together with $C^*_k(X)$, form a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ when $X$ is a proper metric measure space of bounded geometry with no isolated points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09363 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A continuous field of Roe algebras Manuilov, V. Operator Algebras Let $X$ be a metric measure space. A Delone subset $D\subset X$ is a uniformly discrete set coarsely equivalent to $X$. We consider the space $\mathcal D_F$ of controlled Delone subsets of $X$ with an appropriate metric, and show that it, together with $X$ itself, is a compact space. By assigning to each point $D$ of $\mathcal D_F$ (resp., to $X$) the uniform Roe algebra $C^*_u(D)$ (resp., the \u Spakula's version $C_k^*(X)$ of the Roe algebra of $X$) we get a tautological family of $C^*$-algebras. For a sequence $\{D_n\}_{n\in\mathbb N}$ of controlled Delone subsets convergent to $X$ we show that the corresponding uniform Roe algebras $C^*_u(D_n)$, together with $C^*_k(X)$, form a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ when $X$ is a proper metric measure space of bounded geometry with no isolated points. |
| title | A continuous field of Roe algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2312.09363 |