On rigidity of Roe algebras

Roe algebras are C⁎C⁎-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)(X,d). In the special case that X=ΓX=Γ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)(Γ,d) is isomorphic to the crossed product C⁎C⁎-algebra l∞(Γ)⋊rΓl∞(Γ)⋊rΓ.Roe algebras are coarse invariants, meaning that if X and Y   are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum–Connes conjecture, we show that the converse statement is true for a very large classes of spaces. This can be thought of as a ‘C⁎C⁎-rigidity result’: it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly ‘rigid’.As an example of our results, in the group case we have that if Γ and Λ   are finitely generated amenable, hyperbolic, or linear, groups such that the crossed products l∞(Γ)⋊rΓl∞(Γ)⋊rΓ and l∞(Λ)⋊rΛl∞(Λ)⋊rΛ are isomorphic, then Γ and Λ are quasi-isometric.