Diagonal type conditions on group C$^*$-algebras

Let $G$ be a locally compact group with $\mathbf{C}^* (G)$ and $\mathbf{C}^*_r (G)$ its enveloping and reduced C$^*$-algebras respectively. We show that if $\mathbf{C}^*(G)$ is residually finite dimensional, then $G$is maximally almost periodic, and $\mathbf{C}^*_r (G)$ is residually finite dimensional if and only if $G$ is both amenable and maximally almost periodic. Letting $\lambda_G$ be the left regular representation of $G$, we show that a certain quasidiagonality condition on $\{\lambda_G(s):s\in G\}$ implies that $G$ is amenable.