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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.

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    $\begingroup$ +1, as this has bothered me in the past as well. For example, we might naively ask that a *-homomorphism from L^\infty[0,1] to itself should be given by a measure preserving map [0,1] to itself (and I'm certain I've seen this is books). However, it's actually enough that our map should just preserve null sets under inverse image (I think, anyway). $\endgroup$ – Matthew Daws May 4 '10 at 8:14
  • $\begingroup$ Well, there is another explicit version of Gelfand duality for von Neumann algebras in Chapter 7 of Pedersen's book "Von Neumann algebras". In fact, I think this goes back to Von Neumann himself. But there is no mention of morphisms whatsoever (as might be expected from a text from that period). $\endgroup$ – Chris Heunen Apr 20 '11 at 11:26
  • $\begingroup$ @Chris: You probably mean Dixmier, not Pedersen (Pedersen doesn't have such a book and the only book with such a name is by Dixmier). Dixmier describes the second construction, even though he doesn't use the term hyperstonean. $\endgroup$ – Dmitri Pavlov Apr 23 '11 at 9:59
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    $\begingroup$ I found the spelling 'Neumark' disconcerting -- I grew up thinking it was Naimark -- but I eventually discovered that in another MO exchange, Dmitri Pavlov argued very forcefully that 'Naimark' is incorrect and should never be used. May non-Russian readers take note! Reference: mathoverflow.net/questions/25878/… $\endgroup$ – Todd Trimble Sep 8 '11 at 5:44
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    $\begingroup$ I just rolled back an edit that changed Neumark to Naimark. While I personally can't get too upset with people using the old inaccurate transliteration, I think DP has made his case for using "Neumark", and it is his question (not to mention his language...) $\endgroup$ – Yemon Choi Feb 23 '12 at 1:19
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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.

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  • $\begingroup$ Excellent, thanks a lot for the reference! $\endgroup$ – Dmitri Pavlov Oct 17 '11 at 11:26
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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!

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  • $\begingroup$ I knew about this paper and several other papers by Zaharov about hyperstonean spaces before I posted this question, but I couldn't read these papers either. $\endgroup$ – Dmitri Pavlov Oct 10 '11 at 2:48
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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.

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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.

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  • $\begingroup$ Hence it's false that commutative Von Neumann algebra are dual to 'enhanced' measurable space and null-reflecting maps between them? You need to restrict compact stricly localizable and change the maps? $\endgroup$ – mattecapu Feb 16 at 10:41
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    $\begingroup$ @mattecapu: The theorem as stated in my original post is also true (with a different meaning of "measurable maps"), see Remark 5.18 in arxiv.org/abs/2005.05284. $\endgroup$ – Dmitri Pavlov Feb 16 at 14:54
  • $\begingroup$ thanks, I'll look into it. Also thank you for having written such a detailed account of the duality! It's a great piece of work. $\endgroup$ – mattecapu Feb 16 at 17:22
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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.

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    $\begingroup$ Johnstone's book was my original source of motivation for this question. Unfortunately, I was unable to find anything about measurable spaces, hyperstonean spaces, or von Neumann algebras in his book. $\endgroup$ – Dmitri Pavlov Sep 8 '11 at 11:46

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