Proposition 43.4.7.label Let $M, N$ be von Neumann algebras, then every isomorphism $\phi: M \to N$ is normal. In particular, the weak and strong operator topologies on $M$, restricted to bounded subsets, are independent of its representation.
Proof. As $\phi$ is an isomorphism, it is an order isomorphism. By (1) of Definition 43.4.6, $\phi$ is normal. By (3) and (4) of Definition 43.4.6, its restriction to bounded subsets of $M$ is both weak-operator and strong-operator continuous.$\square$
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