TITLE: Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)?
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: A von Neumann algebra is a $C^*$-algebra $A$ that admits a predual,
i.e., a Banach space $A_*$ such that there is an isomorphism $A\to(A_*)^*$.
A morphism of von Neumann algebras is a morphism of $C^*$-algebras $A\to B$ that admits a predual,
i.e., a morphism of Banach spaces $B_*\to A_*$ such that
$(A_*)^*\to (B_*)^*$ is isomorphic to $A\to B$.
The weak topology induced by the predual on $A$ is precisely the ultraweak topology,
and so ultraweakly continuous morphisms are precisely those morphisms that admit a predual.