nLab
split inclusion of von Neumann algebras

Contents

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,NM, N be two von Neumann algebras with MNM \subseteq N. This inclusion is called split if there is a type I-factor FF with

MFN M \subseteq F \subseteq N

Properties

Theorem

The inclusion MNM \subseteq N is split iff there exist faithful normal representations π 1\pi_1 of MM, π 2\pi_2 of NN' such that the map Φ:MNπ 1(M)π 2(N)\Phi: M \vee N' \to \pi_1(M) \otimes \pi_2(N') given by

Φ(mn):=π 1(m)ϕ 2(n) \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)