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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 26th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea An analogue of the [[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 continuous 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. ## Reference * [[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](https://doi.org/10.1016/j.jpaa.2021.106884), [arXiv](https://arxiv.org/abs/2005.05284). <a href="https://ncatlab.org/nlab/revision/Gelfand-type+duality+for+commutative+von+Neumann+algebras/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-type+duality+for+commutative+von+Neumann+algebras">current</a>

   Created:

   Idea

   An analogue of the 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 continuous 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.

   Reference

   • 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.

   v1, current