nLab split inclusion of von Neumann algebras

Idea

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

$$M \subseteq F \subseteq N$$

Properties

Theorem

The inclusion $M \subseteq N$ is split iff there exist faithful normal representations $\pi_1$ of $M$, $\pi_2$ of $N'$ such that the map $\Phi: M \vee N' \to \pi_1(M) \otimes \pi_2(N')$ given by

$$\Phi(mn') := \pi_1(m) \otimes \phi_2(n')$$

extends to a spatial isomorphism, the tensor product used here is the spatial tensor product.

References

Definition 5.4.1 und Lemma 5.4.2 in

• Wollenberg, Baumgärtel: Causal nets of operator algebras. Mathematical aspects of algebraic quantum field theory.