In Popa's preprint https://www.math.ucla.edu/~popa/popa-correspondences.pdf his initial definition of an $N-M$ correspondence between two von Neumann algebras is a Hilbert space $\mathcal{H}$ and a seperately weakly continuous commuting action of two Von Neumann algebras $N$ and $M$ on $\mathcal{H}$. Could somebody clarify what is meant here by weakly continuous as if $N$ is not already seen as a Von Neumann algebra on $\mathcal{H}$ there is no way to define a weak topology on it as it depends on the Hilbert space it is acting on. Does Popa instead mean $\sigma$-weakly continuous?
Von Neumann Correspondence
von-neumann-algebras
Best Answer
Most likely. If you look at the other equivalent definitions he gives, he talks about normal. So yes, it has to be $\sigma$-weak. Most of the time the distinction doesn't matter, because when an element in $N$ is a sot limit of a net in $N$, by Kaplanski's Density Theorem the net can be replaced by a bounded net, and then the convergence is also $\sigma$-sot.