Exactness from Proper Actions

Abstract

In this paper we show that if a discrete group $G$ acts properly isometrically on a discrete space $X$ for which the uniform Roe algebra $C_u^{\ast }(X)$ is exact then $G$ is an exact group. As a corollary, we note that if the action is cocompact then the following are equivalent: The space $X$ has Yu's property A; $C_u^{\ast }(X)$ is exact; $C_u^{\ast }(X)$ is nuclear.

5 pages