# 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 $\mathcal{H}$.

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 $\{\varphi(a_j)\}\subset B$.
• 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