On rigidity of Roe algebras

Spakula, Jan and Willett, Rufus (2013) On rigidity of Roe algebras. Advances in Mathematics, 249, 289-310. (doi:10.1016/j.aim.2013.09.006).

Record type: Article

Abstract

Roe algebras are C⁎-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d). In the special case that X=Γ is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (Γ,d) is isomorphic to the crossed product C⁎-algebra l∞(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⁎-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)⋊rΓ and l∞(Λ)⋊rΛ are isomorphic, then Γ and Λ are quasi-isometric.

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