# Reference request unital normal *-homomorphisms $B(H) \to B(K)$

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.)

• I unfortunately don't have a reference, but I assume you want the word injective in there somewhere. – Josh Keneda Mar 10 at 23:04
• I forgot to write non-zero. Then it’s automatically injective as its kernel is a ultraweakly closed two sided ideal and thus equal $z B(H)$ for some central projection $z$. Only choice is $z=0$ – westerbaan Mar 11 at 8:09
• Ah, my mistake. For some reason I thought you were trying to characterize certain unital normal completely positive maps $\phi: B(H) \rightarrow B(K)$, and I knew something was missing. Turns out I just suck at reading. Carry on. :) – Josh Keneda Mar 11 at 8:35
• See Theorem 5.5. of Chapter IV in Takesaki's book. – MaoWao Mar 11 at 15:37
• Well, $\phi$ is a normal $\ast$-homomorphism onto its image, and the image is a von Neumann algebra. – MaoWao Mar 14 at 12:24