Is a positive map $\varphi$ between von Neumann algebras normal if $x_i \nearrow x$ implies $\varphi(x_i)\nearrow \varphi(x).$

Question

Let $\varphi: M \to N$ be a positive map between von Neumann algebras. Assume that for any increasing net of positive elements $\{x_i\}_{i \in I}$ we have $$\varphi\left(\sup_{i \in I} x_i\right) = \sup_{i \in I} \varphi(x_i).$$

Does it follow that $\varphi$ is a $\sigma$-weakly continuous map? I know this result is true if we work with nets of self-adjoint elements, but I'm wondering of this also holds if we restrict to positive elements. I tried arguing by decomposing a self-adjoint element as a difference of positive elements but the minus sign destroys the possibility of ending up with a proof that works.

I would greatly appreciate a reference or a proof. Anything in Takesaki's first volume can be assumed to be known.

Answer 1

Yes. It is enough to check this for linear functionals, since $\varphi$ is normal if and only if $f\circ\varphi$ is normal for all normal $f\in N^*$.

And for functionals, this is Corollary III.3.11 in Takesaki I. You don't even have to require the condition for all nets of positive elements; increasing nets of projections suffice.

The proof is fairly simple at that stage in Takesaki, since the decomposition in the normal and singular parts has already been established. Kadison-Ringrose (Theorem 7.1.12) take a slightly more convoluted path, but include more equivalent conditions on the way.