State extensions and a Radon-Nikodým theorem for conditional expectations on von Neumann algebras

Carlo Cecchini and Dénes Petz

Vol. 138 (1989), No. 1, 9–24

DOI: 10.2140/pjm.1989.138.9

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 ∘E with respect to E in the sense that E = E_φ0⋅E. If E_ω is only an ω-conditional expectation, then φ₀ ∘ E_ω 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.

Mathematical Subject Classification 2000

Primary: 46L50, 46L50

Secondary: 46L10

Milestones

Received: 20 June 1987

Published: 1 May 1989