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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.

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Ilan Barnea
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    @DmitriPavlov would probably know. – David Roberts Jul 06 '15 at 03:03
  • In fact, he says it in his question: "...subcategory of von Neumann algebras and their morphisms (σ-weakly continuous morphisms of unital C*-algebras)" – David Roberts Jul 06 '15 at 03:08
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    (1) Yes to both questions. (2) Yes, provided that a measurable map is defined as an equivalence class modulo equality on a conegligible set. More details can be found at http://ncatlab.org/nlab/show/measurable+locale, http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820, and http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542 (the last link has a list of further references). – Dmitri Pavlov Jul 06 '15 at 15:56
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    Clarification to (2): equality on a conegligible set is understood in the sense that two measurable maps f and g are identified if and only if the preimages of any measurable set have a negligible symmetric difference. – Dmitri Pavlov Jul 07 '15 at 10:46
  • It looks to me like waitaminute gave a conter-example. If $f$ is the identity map on $I^{\parallel}$ and the map $g$ is the one exchanging $t^+$ with $t^-$, then the preimage of $I^-$ is $I^-$ by $f$ and $I^+$ by $g$. The symmetric difference is thus $I^{\parallel}=I^-\cup I^+$ which is not negligible. – Ilan Barnea Jul 07 '15 at 14:11
  • @IlanBarnea: I^- is not measurable, so this is not a counterexample. – Dmitri Pavlov Jul 07 '15 at 17:12
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    @DmitriPavlov Thanks! What is a good reference for this equivalence of categories? – Ilan Barnea Jul 07 '15 at 17:38
  • @IlanBarnea: Apparently, there is no single unified reference (I wish I had time to write one). I believe that by carefully combing through Fremlin's books you might be able to extract an equivalence between measurable spaces and measure algebras (= measurable locales). The equivalence between measurable locales and commutative von Neumann algebras is quite elementary, almost a one-liner. – Dmitri Pavlov Jul 08 '15 at 13:21
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    @IlanBarnea: And now there is a reference: https://arxiv.org/abs/2005.05284 – Dmitri Pavlov Sep 29 '20 at 03:56
  • If one thinks about measurable functions on the interval [0,1], as a Cartan subalgebra of a semisimple Lie algebra, then measurable automorphisms become similar to the Weil group, which is the group of permutations for gl(n), hence one gets a definition of an infinite permutation group, and may study its representations, Young diagrams, etc – Andrey Radul Mar 29 '23 at 12:50

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The technical part of your problem is resolved in the sadly little-known article "On point realization of $L^\infty$-endomorphisms" by Vesterstrøm and Wils in Math. Scand. 25 (1970) (journal link). By the way, von Neumann algebras are considered from a categorical viewpoint by Guichardet in a paper in Bull. Math. Soc. 90 (1966).

David Roberts
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  • For the paper of Guicharget see http://dmitripavlov.org/scans/guichardet.pdf – David Roberts Jul 06 '15 at 07:58
  • As far as I see they prove something for $\Omega$ a locally compact space and $\nu$ a positive Radon measure. Does this cover all localizable measure spaces? Also, they show that any normal *-homomorphism $$L^{\infty}(\Omega,\nu)\xrightarrow{F} L^{\infty}(\Omega_1,\nu_1)$$ is induced from a map $\Omega\xrightarrow{f} \Omega_1$. Is there any simple way to characterize the maps $f$ that are obtained this way? Does it give an equivalence of categories? – Ilan Barnea Jul 06 '15 at 08:25
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The split interval $I^{\parallel} = \{t^+ : t \in [0,1]\} \cup \{t^- : t \in [0,1]\}$ yields the standard counterexample to the second question (details can be found in Fremlin Vol 3I, section 343, especially 343J, the exercises and the notes and comments at the end of the section). There usually are many maps of the measure space that induce the same morphisms of the measure algebra.

The identity homomorphism of the measure algebra of $I^{\parallel}$ is induced both by the identity map $f$ and by the map $g$ exchanging $t^+$ with $t^-$. Since the measure algebra uniquely determines the associated von Neumann algebra, both these maps induce the identity map of the algebra. Obviously $f(x) \neq g(x)$ for all $x \in I^{\parallel}$, so no identification modulo zero helps here.

Similarly, the measure algebra of $I^{\parallel}$ is canonically isomorphic to the measure algebra of the unit interval, and the two maps $t^{\pm} \mapsto t^-$ and $t^{\pm} \mapsto t^+$ both induce this isomorphism, but they are nowhere equal.

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