Topology of the "normal spectrum" of a commutative von Neumann algebra Kadison and Ringrose define normal states $\omega$ of a von Neumann algebra $A$ as such that $\omega(H_\alpha)\to \omega(H)$ for each monotone increasing net of operators $H_\alpha$ with least upper bound $H$ (definition 7.1.11) 
Let $A$ be a commutative von Neumann algebra and $NS(A)$ be its set of normal characters, and let us endow $NS(A)$ by some natural topology, for example, by the the weak topology generated by elements of $A$. 
Did anybody try to describe the topological properties of $NS(A)$? As far as I understand, usually this space is not compact, but from the construction of the von Neumann envelope it follows that such spaces are "natural covers" for all Hausdorff compact spaces. So I wonder how this picture can be explained from the topological point of view.
 A: If by a normal character you mean a normal morphism of C*-algebras A→C,
then every commutative von Neumann algebra canonically decomposes
as a product of its atomic and diffuse parts,
the atomic part canonically decomposes as a product ∏i∈IC,
and the set of normal characters is canonically isomorphic to I.
In geometric terms, every measurable space is a disjoint union of its atomic and diffuse parts,
and the atomic part is a disjoint union of points, which can be identified with the normal characters of A.
The only natural topology on the set of normal characters is the discrete topology,
in particular the weak topology induced by A is discrete.
To answer the other question, if we take all characters (not necessarily normal),
i.e., the spectrum of the underlying C*-algebra, then the resulting space is hyperstonean.
Furthermore, for the von Neumann envelope of a commutative C*-algebra A
the resulting map of compact Hausdorff spaces C*-Spec(W*-env(A))→C*-Spec(A)
is the hyperstonean cover of C*-Spec(A).
Alternatively, one can invoke the Gelfand-Neumark theorem for commutative von Neumann algebras,
which states in particular that the opposite category of the category of commutative
von Neumann algebras is equivalent to the category of measurable locales,
which in its turn embeds fully faithfully in the category of locales, which is very
similar to the category of topological spaces (and contains Hausdorff topological
spaces as a full subcategory).
Thus the spectrum of a commutative von Neumann algebra is a locale,
and bounded functions on this locale are precisely the elements of the original
commutative von Neumann algebra.
Of course, the argument above proves that the interesting part of this locale
(the one that corresponds to the diffuse part of the original von Neumann algebra)
has no points (but it is highly nontrivial anyway), in particular it is nonspatial,
i.e., does not come from a topological space and thus provides
an example of a pointfree / pointless topological space.
Arguably this fact can be seen as yet another argument for replacing
topological spaces by locales in mathematics.
