A remark on Radon-Nikodým properties of ordered Hilbert spaces

Abstract

We characterize classical Z,2-spaces as well as L2-spaces of W- algebras by appropriate Radon- Nikodym principles. Segal (9) has shown that usual measure theory can be extended reasonably only to measure spaces, whose corresponding L -spaces have the Radon- Nikodym property. The aim of this note is to show that L -spaces obtained via commutative, respectively, noncommutative integration are as well deter- mined as ordered spaces by suitable Radon-Nikodym principles. Order theo- retical characterizations of classical L -spaces were obtained by Sz.-Nagy (11) in the separable and Schaefer (5) in the general case. Noncommutative L - spaces were characterized by Connes (2) and Wittstock and the author (8, 6). Radon-Nikodym properties of these L -spaces were established by Connes (2), Stratila-Zsido (10) and in (7). Suppose s? is a Hilbert space, which is ordered by a self-dual cone ^+ . Let J be the antilinear unitary involution associated with £?+ by (2, Proposition 4.1). Furthermore let Jlh be the ideal center of (%? ,^+) in the sense of Wils—see (1, p. 76)—and Jf — J?k © iJth . Bos (1) has shown that Jf is a commutative W*-algebra with involution x* = JxJ, x G Jf. In particular the cones of operator-positive, respectively, order-positive elements in J! coincide. Definition. (s? ,s?+) has the Radon-Nikodym property if for r\, {€ ?F+ satisfying r < C there exists x G ^#+ such that r — x£.