I have actually a basic quastion about maps between von Neumann algebras.
If I have a map $f:N \to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal and unital?
I suppose that $f$ is unital if $f(1)=1$, and suppose that $f$ is completely positive if $f$ maps any operator with positive spectrum to other with positive spectrum too. But I really don't know.
Many thanks in advance. And apologise for this basic question.