On the invariant translation approximation property for discrete groups

Abstract:

Recently J. Roe considered the question of whether for a discrete group the reduced group $C^*$-algebra $C_r^*(\Gamma )$ is the fixed point algebra of $\{\text {Ad}(\rho _t) \mid t \in \Gamma \}$ acting on the uniform Roe algebra $UC_r^*(\Gamma )$. $\Gamma$ is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact $\Gamma$, is equivalent to the operator space approximation property of $C_r^*(\Gamma )$. We also give a new characterization of exactness and a short proof of the equivalence of exactness of $\Gamma$ and exactness of $C_r^*(\Gamma )$ for discrete groups.