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 ) ⋊ r Γ and ℓ ∞ ( Λ ) ⋊ r Λ are isomorphic if and only if Γ and Λ are bi-Lipschitz equivalent.