# Gelfand-type duality for commutative von Neumann algebras

@article{Pavlov2020GelfandtypeDF,
title={Gelfand-type duality for commutative von Neumann algebras},
author={Dmitri Pavlov},
journal={arXiv: Operator Algebras},
year={2020}
}
• D. Pavlov
• Published 11 May 2020
• Mathematics
• arXiv: Operator Algebras
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.
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#### References

SHOWING 1-10 OF 71 REFERENCES
The point of pointless topology
Introduction. A celebrated reviewer once described a certain paper (in a phrase which never actually saw publication in Mathematical Reviews) as being concerned with the study of "valueless measuresExpand
Fremlin. Measure theory
• Volume 4. Torres Fremlin, Colchester,
• 2003
Volume 4
• 1998
ABCC8 clinical and functional characterization of the Pro1198Leu ABCC8 gene mutation associated with permanent neonatal diabetes mellitus, Takagi 269–273 adipocyte fatty acid binding protein obesityExpand
Fremlin. Real-valued-measurable cardinals
• Set theory of the reals. Israel Mathematical Conference Proceedings
• 1993
Normierte Ringe. Recueil Mathématique
• 1941