nLab hyperstonean space

Definition

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

Here 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.

Hyperstonean duality

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 valuation?s.

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

Hyperstonean cover

Given a compact Hausdorff space $X$, its hyperstonean cover is defined as a continuous map of compact Hausdorff spaces

$$h X \to X$$

that under the Gelfand duality for commutative unital C*-algebras corresponds to the canonical inclusion $C(X)\to C(X)^{**}$ into the double dual.

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

• Masamichi Takesaki, Theory of Operator Algebras, Chapter III, Section 1.

The hyperstonean cover of a compact Hausdorff space is introduced in

• J. Flachsmeyer, Topologization of Boolean algebras, General Topology and Its Relations to Modern Analysis and Algebra IV, Lecture Notes in Mathematics 609 (1977), 81–97. doi.

A bibliography of hyperstonean covers can be found in

• V. K. Zaharov, Hyperstonean cover and second dual extension, Acta Mathematica Hungarica 51:1-2 (1988), 125-149. doi.