nLab commutative von Neumann algebra

Contents

Context

Algebra

Measure and probability theory

Contents

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:

This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.

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.

References

Last revised on May 27, 2022 at 18:39:24. See the history of this page for a list of all contributions to it.