This is not my subject so I apologize if my question is too obvious or understood from other pages.

I read some pages such as

Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras and
von neumann algebras and measurable spaces.
If I understand correctly, there is some correspondence between localizable measure spaces and commutative von Neumann algebras given by
$$(\Omega,\nu)\mapsto L^{\infty}(\Omega,\nu).$$
But I wanted to clarify:

(1) What is the correct notion of a morphism of commutative von Neumann algebras? Is it a normal *-homomorphism? What is the exact definition of *normal*? is it the same as being σ-weakly continuous?

(2) Is it true that the opposite category of the category of commutative von Neumann algebras (with the appropriate class of morphisms) is equivalent to the category of localizable measure spaces and measurable maps? Or do we need to use another type of morphisms between localizable measure spaces?

Any good references on the above will be highly appreciated.

(σ-weakly continuous morphisms of unital C*-algebras)" $\endgroup$ – David Roberts Jul 6 '15 at 3:084more comments