# Automorphism of $W^*$ algebra

Let $\mathfrak{A}$ be von Neumann algebra. It is in particular $C^*$ algebra. Is it true that every $*$-isomorphism of $\mathfrak{A}$ is also $W^*-$isomorphism? (Note that every $*$-isomorphism of $C^*$ algebra is $C^*-$isomorphism).

I'm especially interested in application to concrete von Neumann algebra (weakly closed subalgebra of $B(\mathcal{H})$ - the algebra of bounded operators on Hilbert space $\mathcal{H}$). Is it true that any isomorphism of this algebra is automatically continuous in weak, strong and ultraweak operator topologies on $\mathcal{H}$.

## 1 Answer

Your question raises the point of what is meant by an "isomorphism of von Neumann algebras".

Any C$^*$-isomorphism will of course preserve order, and then it will be normal: if $\varphi:A\to B$ is a $*$-isomorphism and $\{a_j\}\subset A^{\rm sa}$ is a bounded increasing net with least upper bound $a$, then $\varphi(a)$ is the least upper bound for $\{\varphi(a_j)\}\subset B$.

And a normal map is strong and weak-operator continuous on bounded sets. I would say that this is the natural notion of isomorphism of von Neumann algebras. But I stand to be corrected.

• My question involved the weak/strong continuity of $*-$isomorphism of von Neumann algebras. From your answer it follows that $*-$isomorphism is weak/strong continuous on bounded sets. Thus strictly speaking it is not continuous in these topologies. You answer my question but now I wonder what is precisely the definition of $W^*$ isomorphism ($*-$isomorphism continuous in $\sigma-$weak topology?). Feb 24 '14 at 7:37
• what the books say: Blackadar III.2.2.2 p.249 or Theorem 2 p.59, "von Neumann algebras", Jacques Dixmier and a last one that is a little bit different Thm 2.4.23 p.78-79, Bratteli, Robinson vol. 1 Feb 14 '15 at 20:55