The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann algebras;
(2) The category of hyperstonean spaces and hyperstonean maps;
(3) The category of localizable measurable spaces and measurable maps.
[Apparently one more equivalent category can be defined using the language of locales.
Unfortunately, I am not familiar enough with this language to state this variant here.
Any help on this matter will be appreciated.]
While its more famous version for commutative unital C*-algebras is extensively covered in the literature,
I was unable to find any complete references for this particular variant.
The equivalence between (1) and (2) follows from the Gelfand duality theorem for commutative C*-algebras
via restriction to the subcategory of von Neumann algebras and their morphisms (σ-weakly continuous morphisms of unital C*-algebras).
Takesaki in his Theory of Operator Algebras I, Theorem III.1.18,
proves a theorem by Dixmier that compact Hausdorff spaces
corresponding to von Neumann algebras are precisely hyperstonean spaces
(extremally disconnected compact Hausdorff spaces that admit sufficiently many
positive normal measures).
Is there a purely topological characterization of the last condition (existence
of sufficiently many positive normal measures)? Of course we can require that
every meager set is nowhere dense, but this is not enough.
I was unable to find anything about morphisms of hyperstonean spaces in Takesaki's book
or anywhere else.
The only definition of hyperstonean morphism that I know is a continuous map between hyperstonean spaces such that
the map between corresponding von Neumann algebras is σ-weakly continuous.
Is there a purely topological characterization of hyperstonean morphisms?
I suspect that it is enough to require that the preimage of every nowhere dense set is nowhere dense. Is this true?
To pass from (2) to (3) we take symmetric differences of open-closed sets and nowhere dense sets as measurable subsets and nowhere dense sets as null subsets.
Is there any explicit way to pass from (3) to (2) avoiding any kind of spectrum construction (Gelfand, Stone, etc.)?
Any references that cover the above theorem partially or fully and/or answer any of the three questions above will be highly appreciated.
Best Answer
I think we established that the literature is lacking on this question. But I think the "correct" definition of morphisms between hyperstonean spaces can be puzzled together from G. Bezhanishvili's paper "Stone duality and Gleason covers through de Vries duality" (Topology and its Applications 157:1064-1080, 2010), especially section 6.
He proves in detail a duality between the category of complete Boolean algebras and complete Boolean algebra homomorphisms, and the category of extremally disconnected compact Hausdorff spaces and continuous open maps. But commutative von Neumann algebras and normal *-homomorphisms form a full subcategory of the former (via taking projections), which corresponds to the full subcategory of the latter consisting of hyperstonean spaces.
So Gelfand duality really restricts quite cleanly: commutative von Neumann algebras and normal *-homomorphisms are dual to hyperstonean spaces and open continuous maps.