It is known that if $\varphi \colon B(H) \to B(K)$ is a unital non-zero normal $*$-homomorphism (for Hilbert spaces $H$ and $K$), then there exists a Hilbert space $K'$ and a unitary $U' \colon K \to H \otimes K'$ such that $\varphi(a) = U^* (a \otimes 1) U$ for all $a \in B(H)$.
More well-known is the corollary is that any unital normal $*$-isomorphism between type I von Neumann algebras is inner; i.e. of the form $a \mapsto U^*aU$ for some unitary $U$.
I can't find a reference for it though. (I'm only looking for a reference, I already have a proof.)