Both the ultraweak and ultrastrong topologies are intrinsic topologies in the sense that the image of a continuous (unital) $\*$-homomorphism between von Neumann algebras (in either topology) is a von Neumann subalgebra of the target von Neumann algebra, unlike say just norm-continuous $\*$-homomorphisms. One may then speak of the category of von Neumann algebras with morphisms as ultrastrong $\*$-homomorphisms.


Why do we predominantly think of the ultraweak topology as the intrinsic one when presumably there could be many more topologies that are intrinsic in the above sense? I understand that the ultraweak topology is the weak-topology coming from the pre-dual and hence quite natural to study. But is there a guiding logical or category-theoretic principle that tells us to make this choice? 


Thank you.



Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)?
A $\*$-homomorphism between two von Neumann algebras is weak\* to weak\* continuous if and only if it is ultrastrong to ultrastrong continuous. See Proposition III.2.2.2 of [Blackadar's book](https://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf) (which, basically, answers all questions of this type that you might have).


