A UNIFIED APPROACH TO GENERALIZED CONDITIONAL EXPECTATIONS OPERATOR VALUED WEIGHTS AND RADON–NIKODYM DERIVATIVES ON VON NEUMANN ALGEBRAS

Given two von Neumann algebras, ℳ and with , and two normal semifinite faithful weights, φ and ψ on ℳ and respectively, we define a canonical map from {b ∈ ℳ⁺ | φ(b)< ∞} to the set of positive forms on the Hilbert space of the GNS representation of associated to ψ. We show that generalized conditional expectations, operator valued weights and Radon–Nikodym derivatives on von Neumann algebras can be obtained from particular cases of this canonical map.