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](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820) 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.



Reference for the Gelfand duality theorem for commutative von Neumann algebras
As shown in the paper [Gelfand-type duality for commutative von Neumann algebras](https://arxiv.org/abs/2005.05284), the following categories are equivalent.


* The category CSLEMS of compact strictly localizable enhanced measurable spaces,
whose objects are triples $(X,M,N)$, where $X$ is a set, $M$ is a σ-algebra of measurable subsets of $X$,
$N⊂M$ is a σ-ideal of negligible subsets of $X$
such that the additional conditions of compactness (in the sense of Marczewski) and strict localizability are satisfied.
Morphisms $(X,M,N)→(X',M',N')$ are equivalence classes of maps of sets $f:X→X'$ such that $f^\*M'⊂M$ and $f^\*N'⊂N$
(superscript $\*$ denotes preimages)
modulo the equivalence relation of weak equality almost everywhere: $f≈g$ if for all $m∈M'$ the symmetric difference $f^\*m⊕g^\*m$
belongs to $N$.
* The category HStonean of hyperstonean spaces and open maps.
* The category HStoneanLoc of hyperstonean locales and open maps.
* The category MLoc of [measurable locales](https://ncatlab.org/nlab/show/measurable+locale), defined as the full subcategory of the category of locales
consisting of complete Boolean algebras that admit sufficiently many continuous valuations.
* The opposite category CVNA^op of commutative von Neumann algebras,
whose morphisms are normal \*-homomorphisms of algebras in the opposite direction.


The paper contains an extensive discussion with counterexamples why this particular definition of CSLEMS is necessary.
In particular, the choices of strictly localizable vs localizable, weak equality almost everywhere vs equality almost everywhere,
and the property of compactness are all crucial.


Measurable spaces commonly encountered in analysis are typically compact, strictly localizable, and countably separated.
The latter property guarantees that weak equality almost everywhere implies equality almost everywhere.


Notice a curious property of the category MLoc of measurable locales:
it is a full subcategory of the category of [locales](https://ncatlab.org/nlab/show/locale).
Thus, measure theory quite literally is part of (pointfree) general topology.


