Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Autor:	Baudier, Florent P., Braga, Bruno de Mendonça, Farah, Ilijas, Khukhro, Ana, Vignati, Alessandro, Willett, Rufus
Rok vydání:	2021
Předmět:	Mathematics - Operator Algebras Mathematics - Functional Analysis Mathematics - Metric Geometry
Druh dokumentu:	Working Paper
DOI:	10.1007/s00222-022-01140-x
Popis:	We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as \cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we obtain that if $\Gamma$ and $\Lambda$ are finitely generated groups, then the crossed products $\ell_\infty(\Gamma)\rtimes_r\Gamma$ and $ \ell_\infty(\Lambda)\rtimes_r\Lambda$ are isomorphic if and only if $\Gamma$ and $\Lambda$ are bi-Lipschitz equivalent. Comment: 26 pages, second version with revisions
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2106.11391 Zobrazit plný text záznamu View this record from Arxiv