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In many refernce books and papers, type III von Neumann algebra factors were well studied.

If $M$ is a type III von Neumann algebra has non-trivial centers, what is the structure of $M$?

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Given an abelian subalgebra $N$ of a von Neumann algebra $M$, such as the center of $M$, one can always write $N=L^\infty (X)$, for some measure space $X$. Furthermore one can decompose $M$ as a "direct integral" $$ M=\int_X^\oplus M_x\,dx $$ (sort of a continuous direct sum) of von Neumann algebras indexed by $X$. When $N$ is the center of $M$ the factors $M_x$ in this decomposition are simple von Neumann algebras (called "factors" precisely for that reason) [1, 14.2.3]. The moral of the story is that all there remains to be done about von Neumann algebras is to study factors, so no one cares about non-simple von Neumann algebras anymore!

[1] Kadison, Richard V.; Ringrose, John R., Fundamentals of the theory of operator algebras. Vol. 2, Pure and Applied Mathematics, 100-2. San Diego, CA: Academic Press, Inc.

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  • $\begingroup$ What is the form of $M_x$? Does there exist other reference books to give a complete proof of $M=\int_X^\oplus M_x\,dx$? $\endgroup$ Nov 22, 2020 at 22:37
  • $\begingroup$ Part of the trouble in understanding the $M_x$ is that they are only defined a.e. Intuitively speaking, you take a decreasing family of measurable sets $\{E_i\}_i$ of positive measure, whose intersection is $\{x\}$, and look at the limit of subalgebras $1_{E_i}M$. $\endgroup$
    – Ruy
    Nov 22, 2020 at 23:10
  • $\begingroup$ Another reference is Takesaki, vol 1, section 8. Look for Theorem 8.21 (Existence of Disintegration). $\endgroup$
    – Ruy
    Nov 22, 2020 at 23:11
  • $\begingroup$ ,Section 8 ? there are five chapters in vol 1. $\endgroup$ Nov 23, 2020 at 6:56
  • $\begingroup$ Do you still need help locating Theorem 8.21? $\endgroup$
    – Ruy
    Nov 23, 2020 at 15:22

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