Uniform Roe algebras of uniformly locally finite metric spaces are rigid

TL;DR

This paper proves that the structure of uniform Roe algebras completely determines the coarse geometry of the underlying spaces, establishing a rigidity result linking algebraic isomorphisms to geometric equivalences.

Contribution

It establishes that isomorphism of uniform Roe algebras implies coarse equivalence of the underlying metric spaces, and relates Morita equivalence to coarse equivalence.

Findings

Isomorphic uniform Roe algebras imply coarse equivalence.

Coarse equivalence is equivalent to Morita equivalence of uniform Roe algebras.

Crossed products of finitely generated groups are isomorphic iff the groups are bi-Lipschitz equivalent.

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