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