Isomorphism of Von Neumann algebras In general what is it meant to say two Von Neumann Algebras are isomorphic? Is it an injective and surjective *-homomorphism which is also strongly continuous or one which is also normal? I would suppose the correct definition would be normal as that is an intrinsic property.
For example when it is said that any type I factor is isomorphic to $B(H)$ what exactly are the properties of the map?

As noted by westerbaan in the comments, an algebraic *-isomorphism between von Neumann algebras $M$ and $N$ is automatically normal.  This is the typical definition for an isomorphism of von Neumann algebras, and it's the type of morphism used in the classification of Type I factors you mentioned.
You can find more info in Blackadar's Operator Algebras book here (see Section III.2.2).
But the most relevant info is that an algebraic *-isomorphism $\psi$ is always a homeomorphism for the $\sigma$-weak (i.e. ultraweak) and $\sigma$-strong topologies.  If the commutant of $M$ is properly infinite, then the weak and $\sigma$-weak (resp., strong and $\sigma$-strong) topologies coincide on $M$.  So if both $M'$ and $N'$ are properly infinite, then $\psi$ is obviously also a homeomorphism for the weak or strong topologies.  But $\psi$ won't be strongly or weakly continuous in general (i.e. without the assumption about the commutants).
That said, $\psi$ will be a homeomorphism for the strong or weak topologies if you restrict $\psi$ to the unit balls of $M$ and $N$, no matter what their commutants are.  So $\psi: B_1(M) \rightarrow B_1(N)$ is always a homeomorphism for any of the topologies we've talked about.
