Gelfand-type duality for commutative von Neumann algebras

Abstract

We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.

47 pages. Comments and questions are very welcome. v2: Added Theorem 1.2, Proposition 4.59, Remark 5.12. v3: Identical to the journal version except for formatting and style

References in corpus (1)

• Observables I: Stone Spectra

Cited by in corpus (5)

• Abelian von Neumann algebras, measure algebras and L^\infty-spaces
• Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration
• Contextuality and the fundamental theorems of quantum mechanics
• Borel and analytic sets in locales
• Categories of abstract and noncommutative measurable spaces