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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