L-embedded Banach spaces and measure topology

An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known results coincides with the usual measure topology on preduals of finite von Neumann algebras (like L 1 ([0, 1])). Though not numerous, the known properties of this topology suffice to generalize several results on subspaces of L 1 ([0, 1]) to subspaces of arbitrary L-embedded spaces. §1 Introduction This article continues the investigations made in [23, 24] on asymptotically isometric copies of l 1 in preduals of von Neumann algebras and in L-embedded Banach spaces. (For defintions see below.) In [24] it has been proved that, roughly speaking, in the predual of a finite von Neumann algebra the only non-trivial bounded sequences that converge to 0 with respect to the measure topology are essentially those that span l 1 asymptotically; for L 1 (µ), µ a finite measure, this characterization has been known for quite a time [15, Th. 2], [25, Th. 3, Rem. 6bis]. From the point of view of Banach space theory, L-embedded Banach spaces provide a natural frame for preduals of von Neumann algebras. So the starting point of this paper is on the one hand the definition of an abstract measure topology, Definition 3, patterned after the just mentionend characterization and on the other hand the easy but important observation, Theorem 4, that every L-embedded space admits such a topology. Although this topology does not come out easily with its properties-at the time of this writing it is not clear whether it is Hausdorff let alone metrizable or whether addition is continuous-it allows to generalize several results on subspaces of L 1 (µ) to subspaces of arbitrary L-embedded spaces. Thus section §4 of the present paper is titled " Section IV.3 of [13] (partly) revisited ". For example, Theorem 10 generalizes a theorem of Buhvalov-Lozanovskii which describes the link between L-embeddedness and measure topology for subspaces Y of L 1 (µ), µ finite: Y is L-embedded if and only if its unit ball is closed in measure. (Note in passing that this criterion involves only the space Y itself, not its bidual.) Moreover, as a consequence of this, the closedness in measure of the unit ball of Y is a weak substitute for compactness which could …