# Existence of disintegration of von Neumann algebras in Takesaki's book

I'm trying to understand Theorem 8.21 (Existence of disintegration) in Takesaki's book "Theory of Operator Algebras I" which is the main result of writing a von Neumann algebra on a separable Hilbert space as a direct integral of factors.

However, it seems that the proof, using disintegration of representations of C*-algebras, implicitly assumed the Hilbert space on which the von Neumann algebra acts, is already decomposed as a direct integral of a measurable field of Hilbert spaces on some standard Borel space. Therefore I wonder how to establish this assumption, or have I missed the proof of this in the text? Thank you very much!

## 1 Answer

I'm not very familiar with this part of Takesaki's book. I think the answer to your question is Corollary 8.11. It's not immediately obvious to me what Radon measure to use, but here is a possibility: take a countable dense $$\{a_n\}\subset A$$; for each $$a_n$$, get a state $$\varphi_n$$ with $$\varphi(a_n)=1$$. Construct Radon measures $$\mu_n$$ as in the proof of Proposition I.4.5, and form $$\mu=\sum_n2^{-n}\mu_n$$. Note that the quasi-state space is weak$$^*$$-compact.