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.


1 Answer 1


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.

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.

  • $\begingroup$ Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators. $\endgroup$ Dec 29, 2018 at 3:52
  • $\begingroup$ Supremum="least upper bound". If you have an order, you have notion of supremum. $\endgroup$ Dec 29, 2018 at 3:53
  • $\begingroup$ yes, you are right. Thanks again $\endgroup$ Dec 29, 2018 at 4:01

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