L-embedded banach spaces and measure topology

Abstract

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 ( 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 be called "convex sequential compactness", see Corollary 9. We also reprove a result of Godefroy and Li concerning a criterion for L-embedded subspaces which are duals of M-embedded spaces, see Theorem 13. In this vein, that is by substituting arbitrary L-embedded spaces for L 1 (µ), we recover also some results of Godefroy, Kalton, Li [10] in §5. Finally, in §6 it is proved that addition is τ µ -continuous in preduals of von Neumann algebras. §2 Notation, Background: The results are stated for complex scalars. The dual of a Banach space X is denoted by X ′ . B X denotes the unit ball of X. Subspace of a Banach space means norm-closed subspace, bounded always means normbounded. As usual, we consider a Banach space as a subspace of its bidual omitting the canonical embedding. [x n ] denotes the closed linear span of a (finite or infinite) sequence (x n ).Basic properties and definitions which are not explained here can be found in [4] or in [20]-[21] for Banach spaces and in [22,28] for C * -algebras. The standard reference for M-and L-embedded spaces is the monograph [13].

Weak Banach-Saks property and Komlós’ theorem for preduals of JBW*-triples

“…In particular, τ µ -convergent sequences are (norm) bounded. (Note that in [24] the definition of 'to span ℓ 1 almost isometrically' differs slightly from ours but coincides with ours for normalized sequences.) It is not known whether τ µ is Hausdorff but convergent sequences have unique limits (cf.…”

“…1.9.13]). Further, we prove the technical result that addition on a JBW * -predual is jointly sequentially continuous with respect to the abstract measure topology defined in [24]. As a (still technical) consequence this topology is Fréchet-Urysohn on the unit ball.…”

“…The following claim is a consequence of Godefroy's well known way to use the w * -lower semicontinuity of the norm in L-embedded Banach spaces ( [13], [15, IV.2.2]) (as it has been used, for example, throughout [24]). The proof is inserted here for the reader's convenience.…”

“…The theorem is immediate from the splitting property shown in [23, Thm. 6.1], from the weak Banach-Saks property Corollary 2.5 and from[24,§6, Observation] or[26, p. 637].…”

Komlós’ theorem and the fixed point property for affine mappings

“…Note that L 1 (M, τ ) is a L-embedded Banach space and the measure topology is an abstract measure topology in the sense of [17]. Thus, equality ( * ) given in the proof of Lemma 3.2 can be generalized to the frame of L 1 (M, τ ) and the measure topology (see [11,16,17]) and Komlós' condition is extended in [19,Proposition 3.11]. As a consequence of Theorem 3.3 we can conclude: In case that C is closed in measure, as the close unit ball, T has a fixed point if S(T ) < 2.…”