State extensions and a Radon-Nikodým theorem for conditional expectations on von Neumann algebras
Carlo Cecchini, Dėnes Petz · Pacific Journal of Mathematics · 1989
Let M be a von Neumann algebra with a von Neumann subalgebra MQ.If £ is a conditional expectation (i.e., projection of norm one) from M into Λ/ o , then any faithful normal state φ 0 admits a natural extension φo o E with respect to E in the sense that E = E φQ .E .If E ω is only an ω-conditional expectation, then φ 0 o E ω is not always an extension of φ 0 .This paper is devoted to the construction of an extension φo of ψo generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ωconditional expectation under suitable majorization condition.