Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$
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| Format: | Preprint |
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2021
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| _version_ | 1866909675079860224 |
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| author | Kato, Tsuyoshi Kishimoto, Daisuke Tsutaya, Mitsunobu |
| author_facet | Kato, Tsuyoshi Kishimoto, Daisuke Tsutaya, Mitsunobu |
| contents | We study the homotopy type of the space of the unitary group $\U_1(C^\ast_u(|\mathbb{Z}^n|))$ of the uniform Roe algebra $C^\ast_u(|\mathbb{Z}^n|)$ of $\mathbb{Z}^n$. We show that the stabilizing map $\U_1(C^\ast_u(|\mathbb{Z}^n|))\to\U_\infty(C^\ast_u(|\mathbb{Z}^n|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $\U_1(C^\ast_u(|\mathbb{Z}^n|))$, which is the product of the unitary group $\U_1(C^\ast(|\mathbb{Z}^n|))$ (having the homotopy type of $\U_\infty(\mathbb{C})$ or $\mathbb{Z}\times B\U_\infty(\mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^\ast(|\mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2102_00606 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$ Kato, Tsuyoshi Kishimoto, Daisuke Tsutaya, Mitsunobu Algebraic Topology K-Theory and Homology Operator Algebras We study the homotopy type of the space of the unitary group $\U_1(C^\ast_u(|\mathbb{Z}^n|))$ of the uniform Roe algebra $C^\ast_u(|\mathbb{Z}^n|)$ of $\mathbb{Z}^n$. We show that the stabilizing map $\U_1(C^\ast_u(|\mathbb{Z}^n|))\to\U_\infty(C^\ast_u(|\mathbb{Z}^n|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $\U_1(C^\ast_u(|\mathbb{Z}^n|))$, which is the product of the unitary group $\U_1(C^\ast(|\mathbb{Z}^n|))$ (having the homotopy type of $\U_\infty(\mathbb{C})$ or $\mathbb{Z}\times B\U_\infty(\mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^\ast(|\mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces. |
| title | Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$ |
| topic | Algebraic Topology K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2102.00606 |