$^*$-endomorphisms are completely positive

Question

Let $\mathfrak{A}$ be a $C^*$-algebra and $\pi : \mathfrak{A} \rightarrow B(\mathcal{H})$ a $^*$-homomorphism.

It is immediately verified that $\pi$ is positive since $\pi(a^*a) = \pi(a)^*\pi(a)$. But how to verify that $\pi$ is completely positive? Is this a direct consequence of the Stinespring theorem?

EDIT:

Maybe it's just because you can use the multiplicative property on matrices as well, that is, considering

\begin{align} \pi : M_n(\mathfrak{A}) &\longrightarrow M_n(B(\mathcal{H})) \end{align}

then it holds $\pi((a_{ij})^*(a_{ij})) = \pi((a_{ij}))^*\pi((a_{ij}))$ (where $(a_{ij}) \in M_n(\mathfrak{A})$) and the latter is a positive element of $M_n(B(\mathcal{H}))$..

Answer 1

Your edit is a correct proof. It suffices to verify two things:

1. Every $*$-homomorphism between $C^*$-algebras is a positive map, and
2. If $\pi:A\to B$ is a $*$-homomorphism, then the map $\pi_n:M_n(A)\to M_n(B)$ given by $\pi_n(a_{ij})=(\pi a_{ij})$ is a $*$-homomorphism.