Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Abstract

We show that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras, Cu∗(X) and Cu∗(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 Cu∗(X) and Cu∗(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.