Abstract: Coarse geometry is the study of large-scale properties of metric spaces. Two metric spaces are coarsely equivalent if their "large-scale geometries" agree. The uniform Roe algebra $C^*_u(X)$ is a norm-closed algebra of bounded linear operators on the Hilbert space $\ell_2(X)$. It is the algebra of all bounded linear operators on $\ell_2(X)$ that can be uniformly approximated by operators of "finite propagation". The uniform Roe algebra is a coarse invariant of the space $X$. It includes $\ell_\infty(X)$ (as the algebra of all operators of zero propagation) and the algebra of compact operators. The uniform Roe corona $Q^*_u(X)$ is obtained by modding out the compact operators from $C^*_u(X)$. After introducing the basics of coarse spaces and uniform Roe algebras, we will study implications (or lack thereof) between the following three assertions and their variants:
|IF.A.1. ||The spaces $X$ and $Y$ are (bijectively) coarsely equivalent. |
|IF.A.2. ||The uniform Roe algebras of $C^*_u(X)$ and $C^*_u(Y)$ are isomorphic. |
|IF.A.3. ||The uniform Roe coronas $Q^*_u(X)$ and $Q^*_u(Y)$ are isomorphic. |
Under some additional assumptions on $X$ and $Y$ (the uniform local finiteness and a weakening of Yu’s property A — A stands for "amenability"), IF.A.2 implies IF.A.1. The implication from IF.A.3 to IF.A.2, even for uniformly locally finite spaces with property A, nontrivially (and possibly necessarily) involves set theory. This talk will be based on a joint work with B.M. Braga and A. Vignati.