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Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra. Moreover, a morphism of commutative von Neumann algebras induces a continuous morphism of the corresponding complete Boolean algebras.

Thus we have a fully faithful functor F from the category of commutative von Neumann algebras to the category of complete Boolean algebras and their continuous morphisms.

The category of complete Boolean algebras and their continuous morphisms is a full subcategory of the opposite category of the category of locales.

Thus the functor F can be seen as implementing the Gelfand-Neumark duality for commutative von Neumann algebras. However, to obtain a satisfactory statement of the duality we still need to characterize in topological terms objects in the essential image of F, which we call measurable spaces (or locales, think of a localic version of point-set measurable spaces).

What additional topological conditions do we need to impose on a complete Boolean algebra to ensure that it is the algebra of projections of some von Neumann algebra, i.e., a measurable space?

It is relatively easy to pin down non-topological conditions. For example, a complete Boolean algebra comes from a von Neumann algebra if and only if it admits sufficiently many normal positive measures.

The reason for requiring additional conditions to be topological is that the resulting definition of a measurable space should be easy to relate to other parts of general topology.

For example, consider the forgetful functor that sends a commutative von Neumann algebra to its underlying C*-algebra. Applying the Gelfand-Neumark duality to both sides we obtain the forgetful functor from the category of measurable spaces to the category of compact regular locales (or compact Hausdorff spaces, if we have the axiom of choice). A topological definition of a measurable space should allow for an explicit description of this forgetful functor in terms of open sets. Other potential applications include functors that send a locale (or a topological space) to its underlying measurable space, or a smooth manifold to its underlying measurable space. More speculatively, one could use this definition to replace ad hoc techniques of classical point-set measure theory with standard tools of general topology.

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Can you give any examples of complete Boolean algebras that cannot occur as the projection algebra of a commutative von Neumann algebra? (Say, using your "non-topological condition"?) –  Manny Reyes Jul 26 '11 at 22:21
@Manny: The Boolean algebra of clopen sets of any compact regular extremally disconnected space (i.e., a stonean space) is a complete Boolean algebra, which does not come from a von Neumann algebra unless the original space was hyperstonean. In a hyperstonean space meager sets are rare (nowhere dense), which is usually not the case for stonean spaces. In fact, every stonean space canonically splits as a disjoint union of a hyperstonean space, a space that contains a dense meager set, and a space where every meager set is rare and the support of every measure is rare. –  Dmitri Pavlov Jul 28 '11 at 12:16
This is probably not relevant, but what does it mean to admit ‘sufficiently many’ positive normal measures? (I assume that ‘normal’ here means that the rule that $\lim_n \mu(A_n) = 0$ when $A_n \searrow_n \varnothing$ applies for any net $A$, not just for an $\omega$-sequence.) –  Toby Bartels Oct 8 '11 at 6:32
@Toby: Here “sufficiently many positive normal measures” means that for any x≠0 we can find a positive normal measure μ such that μ(x)=1. Equivalently we can say that the supremum of the supports of all normal positive measures equals 1. By a normal positive measure here I mean an additive map from the boolean algebra to the positive reals that preserves suprema of arbitrary sets. Your condition is equivalent to the preservation of suprema of arbitrary sets. –  Dmitri Pavlov Oct 8 '11 at 12:41
@Toby: One reference for this notion is Takesaki's Theory of Operator Algebras I, Definition III.1.14, Theorem III.1.17, and Theorem III.1.18. –  Dmitri Pavlov Oct 8 '11 at 12:43
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