Uniform Roe algebras of uniformly locally finite metric spaces are rigid
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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 $ell_infty(Gamma) times_rGamma$ and $ ell_infty(Lambda) times_rLambda$ are isomorphic if and only if $Gamma$ and $Lambda$ are bi-Lipschitz equivalent.