nLab commutative von Neumann algebra

Idea

A von Neumann algebra that is commutative.

Properties

The category of commutative von Neumann algebras is a full subcategory of that of von Neumann algebras and has many special properties.

Duality

The following five categories are equivalent:

• the opposite category of commutative von Neumann algebras;
• compact strictly localizable enhanced measurable spaces;
• measurable locales;
• hyperstonean locales;
• 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.

See Gelfand-type duality for commutative von Neumann algebras for more information.

Properties

From https://mathoverflow.net/questions/384346/is-the-opposite-category-of-commutative-von-neumann-algebras-a-topos/384357#384357:

The opposite category of commutative von Neumann algebras is not a topos because categorical products with a fixed object do not always preserve small colimits. See Theorem 6.4 in Andre Kornell’s Quantum Collections.

Closed monoidal structure

The opposite category of commutative von Neumann algebras admits a non-cartesian closed monoidal structure, where the monoidal product corresponds to the spatial product of measurable spaces.

Related concepts

• von Neumann algebra

• Hilbert W*-module

• W*-representation

• duality between geometry and algebra

References

• Jacques Dixmier, Sur certains espaces considérés par M. H. Stone.

  Summa Brasiliensis Mathematicae 2 (1951), 151–182. PDF.

• Irving E. Segal, Equivalences of measure spaces.

  American Journal of Mathematics 73:2 (1951), 275–313. doi:10.2307/2372178.

• Dmitri Pavlov, Gelfand-type duality for commutative von Neumann algebras.

  2005.05284