Contents

# Contents

## Definition

A hyperstonean space (French: espace hyperstonien) is a Stonean space such that the union of supports of all normal measures is everywhere dense.

A normal measure is a Radon measure that vanishes on nowhere dense subsets.

## Properties

The support of a normal measure is a clopen subset.

In a hyperstonean space, all meager subsets are nowhere dense and the support of any normal measure is a clopen subset.

Every Stonean space decomposes as a coproduct of three clopen subsets $E_1$, $E_2$, and $E_3$ with the following properties:

• $E_1$ has an everywhere dense meager subset. The only normal measure on $E_1$ is the zero measure.

• $E_2$ is a hyperstonean spaces.

• Every meager subset of $E_3$ is nowhere dense and every Radon measure has a nowhere dense support. The only normal measure on $E_3$ is the zero measure.

By Gelfand-type duality for commutative von Neumann algebras?, the category of hyperstonean spaces and open continuous maps is equivalent to the opposite category of localizable Boolean algebras?, the category of measurable locales, and the opposite category of commutative von Neumann algebras.

## Hyperstonean locales

Hyperstonean locales can be defined as Stonean locales that admit sufficiently many normal valuations?.

Assuming the axiom of choice, Stonean locales are spatial?, so the category of hyperstonean locales is equivalent to the category of hyperstonean spaces.

## References

Hyperstonean spaces were introduced and studied by Jacques Dixmier:

• Jacques Dixmier, Sur certains espaces considérés par M. H. Stone. Summa Brasiliensis Mathematicae 2, (1951), 151–182. PDF

An expository account is given by Masamichi Takesaki in

Last revised on July 3, 2021 at 11:35:58. See the history of this page for a list of all contributions to it.