Rigidity of Uniform Roe Algebras

Title: Rigidity of Uniform Roe Algebras

Speaker: Bruno de Mendonça Braga (U. of Virgina)

Abstract:  Given a uniformly locally finite metric space $X$ (also known as a metric space with bounded geometry), one can define a C*-algebra of bounded operators on $\ell_2(X)$ called the uniform Roe algebra of $X$, denoted by $C^*_u(X$). The quintessential examples of uniformly locally finite metric spaces are the Cayley graphs of finitely generated groups endowed with the shortest path metric. Uniform Roe algebras were introduced by John Roe in the context of index theory of elliptical operators on noncompact manifolds and they capture many of the large-scale (i.e., coarse) geometric properties of $X$. The rigidity problem for uniform Roe algebras is the problem of whether such algebras remember all the large-scale aspects of their base metric spaces. Precisely, if $C^*_u(X)$ and $C^*_u(Y)$ are isomorphic as C*-algebras, does it follow that X and Y are coarsely equivalent? This problem has recently been solved by myself together with F. Baudier, I. Farah, A. Khukhro, A. Vignati, and R. Willett. In this talk, I will give an overview of this rigidity problem and explain the main steps involved in its solution.

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