Gelfand-type duality for commutative von Neumann algebras

Dmitri Pavlov

doi:10.1016/j.jpaa.2021.106884

Gelfand-type duality for commutative von Neumann algebras

Dmitri Pavlov

Mathematics and Statistics

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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. This paper is also available as arXiv:2005.05284v3.

Original language	English
Article number	106884
Journal	Journal of Pure and Applied Algebra
Volume	226
Issue number	4
DOIs	https://doi.org/10.1016/j.jpaa.2021.106884
State	Published - Apr 2022

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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. This paper is also available as arXiv:2005.05284v3.",

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AB - 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. This paper is also available as arXiv:2005.05284v3.

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Gelfand-type duality for commutative von Neumann algebras

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