The split property for inclusions of von Neumann algebras was first introduced by Detlev Buchholz in the study of the AQFT approach to quantum field theory, but has become a much used concept in the mathematical structure theory as well.

Definition

Let $M,N$ be two von Neumann algebras with $M\subseteq N$. This inclusion is called split if there is a type I-factor $F$ with

$$M\subseteq F\subseteq N$$M \subseteq F \subseteq N

Properties

Theorem

The inclusion $M\subseteq N$ is split iff there exist faithful normal representations ${\pi}_{1}$ of $M$, ${\pi}_{2}$ of $N\prime $ such that the map $\Phi :M\vee N\prime \to {\pi}_{1}(M)\otimes {\pi}_{2}(N\prime )$ given by