nLab Gelfand-type duality for commutative von Neumann algebras

Idea

An analogue of Gelfand duality for commutative von Neumann algebras.

Statement

The following five categories are equivalent.

• The opposite category of commutative von Neumann algebras and normal \*-homomorphisms.

• The category of measurable locales, which is a full subcategory of the category of locales.

• The category of hyperstonean locales and open maps of locales.

• The category of hyperstonean spaces and open maps.

• The category of compact strictly localizable enhanced measurable spaces and equivalence classes of measurable negligibility-reflecting maps modulo the equivalence relation of weak equality almost everywhere.

References

• Dmitri Pavlov, Gelfand-type duality for commutative von Neumann algebras, Journal of Pure and Applied Algebra 226:4 (2022), 106884, 1–53. doi:10.1016/j.jpaa.2021.106884, arXiv.