Let $G$ be a locally compact group. In order to make things simpler, let us say it is discrete. Let $\pi$ and $\rho$ be unitary representations of $G$ on Hilbert spaces $H_\pi$ and $H_\rho$ respectively. Let $C^*_\pi(G)$, resp. $C^*_\rho(G)$ be the corresponding group $C^*$-algebras, i.e. the $C^*$-algebras generated by $\{\pi(g): g\in G\}\subseteq B(H_\pi)$, and analogously for $\rho$.
It seems to be a folklore fact that the following two facts are equivalent
- $\rho\prec \pi$, i.e. $\rho$ is weakly contained in $\pi$;
- the map $\pi(g)\to\rho(g)$ extends to a ${}^*$-homomorphism from $C^*_\pi(G)$ onto $C^*_\rho(G)$; or equivalently, for every $x\in \mathbb{C}G$, $\|\pi(x)\|_{B(H_\pi)}\geq \|\rho(x)\|_{B(H_\rho)}$.
It looks quite clear on the first sight, however when trying to write down a precise proof, I get stuck.
I would be grateful for either writing down the proof or providing a reference where this is proved exactly as stated.
The result is Theorem 7 in de la Harpe's paper "On simplicity of Reduced $C^*$-Algebras of Groups" (here is a link). The proof is skimmed there, and the details are not hard to fill in.