# 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.

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For $A=L^\infty[0,1]$ don't you get that $NS(A)$ is empty? –  Matthew Daws Apr 5 '12 at 17:20
Yes, I have corrected: USUALLY NS(A) is not compact. Thank you. –  Sergei Akbarov Apr 5 '12 at 18:21