The functoriality of group C* algebra structure

Actually, I'm not quite sure where this is explained completely basically in the literature. So maybe here's a hint, at least for Q1. The definition of $C^*(G)$ is that it's the completion of L^1(G) with respect to the biggest C*-norm. So if $phi:Grightarrow H$ is a continuous group homo, we immediately get a *-homomorphism $ell^1(G) rightarrow ell^1(H)$, and so by inclusion, a *-homo $ell^1(G) rightarrow C^*(H)$. But this defines some C*-norm on $ell^1(G)$, so the norm on $C^*(G)$ must dominate this, and hence we get a continuous extension to $C^*(G) rightarrow C^*(H)$.

For Q2, find a proof in the literature (I think this goes back to Godemont?) that $C^*_r(G) = C^*(G)$ if and only if the left-regular representation contains the trivial representation, if and only if G is amenable. Put another way, if G is not amenable, then the trivial homomorphism $Grightarrow{1}$ doesn't induce a map $C^*_r(G)rightarrow C^*({1}) = mathbb C$.