Given two von Neumann algebras, ℳ and [Formula: see text] with [Formula: see text], and two normal semifinite faithful weights, φ and ψ on ℳ and [Formula: see text] 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 [Formula: see text] 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.

Let N ⊂ M be von Neumann algebras and Eω: M → N an ω-conditional expectation mapping. For a state ψ of N an extension \̃gyEω of ψ with respect to Eω is described. The relation Eω ~ Eϑ defined to hold if \̃gyEω = \̃gyEϑ for every ψ is an equivalence relation. The family of equivalence classes possesses an affine structure and shows analogy with the normal state space of a von Neumann algebra.

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.

Conditional expectations play an important role in classical probability theory. In the general context of von Neumann algebras they were impliciteiy used by von Neumann [41, Chap. II] and by Dixmier [14]. Nakamura and Turumaru [27] and Umegaki [36-391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von Neumann (or C*-) algebras and established many properties of these objects especially in the context of von Neumann algebras with a finite trace. Their starting point was the characterization, given by Moy [26], of the classical conditional expectations as operators on spaces of measurable functions. Tomiyama showed [33 ] that conditional expectations, in the sense of the above mentioned authors, can be characterized as norm one projection in C*algebras. The importance of norm one projection in the classification problem of von Neumann algebras was recognized by Hakeda and Tomiyama [23] and subsequent research on this argument confirmed the usefulness of these objects. This line of thought culminated in the fundamental work of Connes [ 121 in which approximately finite von 245 0022.1236/82/020245-29$02.00/O

Let G be a locally compact, noncompact, unimodular group.For xeG, we denote by L x , the left translation operator defined on L 2 (G) by LJ{y) = /(or 1 ?/).We let Sf 2 {G) be the closure, in the weak operator topology, of the algebra generated by the operators {L x \ xβG}.For fe L P (G), 1 ^ p 2, we let L f be the closed operator in L 2 (G), defined by L f g = f*g, for geL 1 (G) Π L 2 (G).We prove, under a natural hypothesis on G, that for every 1 < p < 2, there exists a projection Pe£?i(G), PφO, with the property that if feL p (G), and PL/ = L fi then / = 0. Thus P is a projection of uniqueness in the sense that the only element /e L p , such that the range of L f is contained in the range of P is the zero element.Another way to express this result is the following: There exists a nontrivial closed subspace of L 2 (G), invariant under right translations and which contains no nonzero element of L P (G).