1
$\begingroup$

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 $M$?

$\endgroup$
0

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
15
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.