Abstract
A simple approach to non-commutative integration for weights is described, following the lines of [7] i.e., using a natural upper integral (which is in fact an integral) and interpolation. If [Formula: see text] is a von Neumann algebra on the Hilbert space H and φ is a faithful normal semifinite weight on [Formula: see text], the space D of all φ-bounded vectors in H is contained in the domain of every closed positive form coming from a positive self-adjoint operator T affiliated to [Formula: see text] with finite upper integral [Formula: see text]. The (classes of) linear combinations of such forms constitute [Formula: see text]. In an obvious sense, [Formula: see text] consists of forms, too (bounded ones). [Formula: see text] is the complex interpolation space [Formula: see text]. It is checked that [Formula: see text] is isometrically isomorphic to Vp in [10], so [Formula: see text] is what it ought to be.
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