# Do all *-isomorphisms between von Neumann algebras preserve strong operator topology?

Do all $$*$$-isomorphisms between von Neumann algebras preserve the strong operator topology?

Seems clearly true for $$*$$-isomorphisms with a unitary implementation, but I don't see the answer for other cases ... perhaps there is an easy argument from the fact that von Neumann algebras are closed in this topology, but I've spent a while looking for one and don't see it.

No. Take $$M$$ to be any II$$_1$$-factor. Let $$\pi:M\to B(H)$$ be an irreducible representation (it exists because you can do GNS of a pure state). As $$M$$ is simple (as a C$$^*$$-algebra!), $$\pi$$ is injective. And $$\pi(M)$$ is dense in $$B(H)$$, but it cannot be everything.
So $$\pi:M\to\pi(M)$$ is a $$*$$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.
As mentioned in the comments, if $$M\subset B(H)$$ and $$N\subset B(K)$$ are von Neumann algebras (in the usual "double commutant" sense) then a $$*$$-isomorphism $$\pi:M\to N$$ is sot-continuous on bounded sets by passing through normality.
• But in this case $\pi$ is not a $\ast$-isomorphism since it is not surjective. Nov 29, 2018 at 12:17
• I am also not convinced since $\ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $\pi$. Perhaps I am confused about that. Nov 29, 2018 at 12:18
• @Adrián: of course, I can take $\pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $\iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra. Nov 29, 2018 at 15:09
• No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me). Nov 29, 2018 at 15:33