A thorough study of type III von Neumann algebras

Question

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 $M$?

Answer 1

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.