Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Abstract

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 ${\ell }_{\infty }(\Gamma ){⋊}_r\Gamma$ and ${\ell }_{\infty }(\Lambda ){⋊}_r\Lambda$ are isomorphic if and only if $\Gamma$ and $\Lambda$ are bi-Lipschitz equivalent.

26 pages, second version with revisions

References in corpus (1)

• Some notes on property A

Cited by in corpus (4)

• Corona Rigidity
• C*-rigidity of bounded geometry metric spaces
• A quantization of coarse spaces and uniform Roe algebras
• K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations