State extensions and a Radon-Nikodym theorem for conditional expectations on Von Neumann algebras

Abstract

Let M be a von Neumann algebra with a von Neumann subalgebra M0. If E is a conditional expectation (i.e., projection of norm one) from M into M0, then any faithful normal state φ0 admits a natural extension φ0 {ring-operator} E with respect to E in the sense that E = Eφ0·E• If Eω is only an ω-conditional expectation, then φ0 {ring-operator} Eω is not always an extension of φ0. This paper is devoted to the construction of an extension φ0 of φ0 generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ω-conditional expectation under suitable majorization condition. © 1989 by Pacific Journal of Mathematics.