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.

Created on June 9, 2010 10:46:53
by Tim van Beek
(192.76.162.8)