Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras, C^*_u(X) and C^*_u(Y) , are isomorphic as C^* -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 C^*_u(X) and C^*_u(Y) . As an application, we obtain that if Γ and Λ are finitely generated groups, then the crossed products ℓ _∞ (Γ )⋊ _rΓ and ℓ _∞ (Λ )⋊ _rΛ are isomorphic if and only if Γ and Λ are bi-Lipschitz equivalent.