Gelfandtype 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 measuretheoretic counterpart of the Gelfand duality between commutative unital C*algebras and compact Hausdorff topological spaces.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.05284
 Bibcode:
 2020arXiv200505284P
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Category Theory;
 Mathematics  Functional Analysis;
 Mathematics  General Topology;
 46L10 (Primary) 54G05;
 54B30;
 06D22;
 18F70;
 06E15;
 28A51;
 28A60;
 28A20 (Secondary)
 EPrint:
 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