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On Rigidity of Roe algebras

Jan Spakula, Rufus Willett
published 2011-10-07

Roe algebras are C*-algebras built using large-scale aspects of a metric space X. In the special case that X is a finitely generated group G, endowed with a word metric, the simplest Roe algebra associated to G is isomorphic to the reduced crossed product C*-algebra of bounded functions on G, by G. Roe algebras are coarse invariants, in the sense 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 ask if there is a converse to the above statement, that is, if X and Y are metric spaces with isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for very large classes of spaces the answer to this question, and some related questions, is yes. 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.

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