# About “names” of von Neuman algebra morphisms

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.

Normal means that $$f$$ respect suprema of bounded nets of selfadjoints. That is, if $$\{a_j\}\subset M$$ and $$a=\sup a_j$$, then $$f(a)=\sup f(a_j)$$. This is the same as saying that $$f$$ is sot-sot continuous.
Positive means that $$f(x)\geq0$$ if $$x\geq0$$. Note that $$x\geq0$$ not only means that $$\sigma(x)\subset[0,\infty)$$ but also that $$x$$ is selfadjoint.
Completely positive means that $$f^{(n)}=f\otimes I_n:M\otimes M_n(\mathbb C)\to N\otimes M_n(\mathbb C)$$ is positive for all $$n\in\mathbb N$$. That is if $$X\in M_n(M)$$ is positive, then $$[f(X_{kj})]_{k,j}\in M_n(N)$$ is positive.