On the invariant translation approximation property for discrete groups

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 $\{$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.