Noncommutative Geometry and Topology in Prague

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Coarse geometry is the study of metric spaces when one forgets about the small scale structure and focuses only on large scales. Objects of interests are coarse spaces, while relevant maps between them are coarse equivalences, required, in short, only to send close points to close points, and far points to far points. Typical examples are finitely generated groups with word metrics, and discretizations of non-discrete spaces such as Riemannian manifolds. To a coarse space one associates, after Roe, several C*-algebras aiming to detect algebraically the geometrical behaviour of the spaces. Chief among the is the uniform Roe algebra of X, C_u^*(X), key for its applications to index theory, mathematical physics, and recently topological phases of matter.

It is natural to ask how much of the underlying coarse geometry can be detected by these C*-algebras. We answer this question, by showing that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras are isomorphic, then X and Y are coarsely equivalent.