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 $\ell_\infty(\Gamma)\rtimes_r\Gamma$ and $ \ell_\infty(\Lambda)\rtimes_r\Lambda$ are isomorphic if and only if $\Gamma$ and $\Lambda$ are bi-Lipschitz equivalent.