Reference for the Gelfand duality theorem for commutative von Neumann algebras

Question

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.

Answer 1

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.

Answer 2

As shown in the paper Gelfand-type duality for commutative von Neumann algebras, 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, 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. Thus, measure theory quite literally is part of (pointfree) general topology.

Answer 3

As far as I know the only purely point-set theoretic (avoiding measure theory) description of the hyperstonian cover (and morphisms thereof) was done by Zakharov in terms of so called Kelley ideals:

V. K. Zaharov, Hyperstonean cover and second dual extension, Acta Mathematica Hungarica Volume 51, Numbers 1-2, 125-149

I tried to read that paper but I failed. Good luck!

Answer 4

This is only an historical comment. As far as I know the equivalence between (1) and (2) is not an easy consequence of Gelfand-Neu(ai)mark theorem. One implication (I do not remember which one) was proved by Dixmier and the other one by Grothendiek. I am quite sure that Dixmier used explicitely the word von Neumann algebra. I have never read Grothendieck's paper but it is likely that he did not use this name and he just proved one of the two implications of the following theorem: $C(K)$ is a dual Banach space iff $K$ is hyperstonean.

J.Dixmier, Sur certains espaces consideres par M.H. Stone, Summa Brasil. Math. 2, 151-182 (1951)

Grothendieck, Sur les applications lineaire faiblement compactes d'espace du type C(K), Canad. J. Math. 5, (1953) 129-173.

Answer 5

Try the book of Peter T. Johnstone, "Stone Spaces" (Cambridge University Press, 1982). He works in the language of locales, which is unfortunately completely alien to me. Hope it helps.