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!
Best 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.