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.
You are right about unital.
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 on the unit ball.
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.