Uniformly locally finite metric spaces with isomorphic Roe algebras are coarsely equivalent, and the outer automorphism group of the Roe algebra is canonically isomorphic to the group of coarse equivalences.
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C*-rigidity of bounded geometry metric spaces
Uniformly locally finite metric spaces with isomorphic Roe algebras are coarsely equivalent, and the outer automorphism group of the Roe algebra is canonically isomorphic to the group of coarse equivalences.