# A thorough study of type III von Neumann algebras

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 $$M$$?

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.

• What is the form of $M_x$? Does there exist other reference books to give a complete proof of $M=\int_X^\oplus M_x\,dx$? Nov 22, 2020 at 22:37
• 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 $1_{E_i}M$.
– Ruy
Nov 22, 2020 at 23:10
• Another reference is Takesaki, vol 1, section 8. Look for Theorem 8.21 (Existence of Disintegration).
– Ruy
Nov 22, 2020 at 23:11
• ，Section 8 ? there are five chapters in vol 1. Nov 23, 2020 at 6:56
• Do you still need help locating Theorem 8.21?
– Ruy
Nov 23, 2020 at 15:22