Non-commutative subsequence principles

Randrianantoanina, Narcisse

doi:10.1007/s00209-003-0560-9

Cited by 8 publications

(3 citation statements)

References 26 publications

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“…For the special case that τ (1) < ∞ and E = L 1 , this result is proved in [Ran1] using somewhat different techniques.…”

Section: For the Measure Topology And There Exist Sequences {Pmentioning

confidence: 99%

“…The preceding definition was introduced by Randrianantoanina [Ran2] (see also [Ran1]), although related notions had been earlier considered in [CS]. If E = L 1 [0, ∞), then the well known criterion of Akemann [Ta] (see also [RX]) asserts that the E-equiintegrable subsets of L 1 (M, τ ) are precisely those which are relatively weakly compact.…”

Section: P-banach-saks Propertiesmentioning

confidence: 99%

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Banach–Saks properties in symmetric spaces of measurable operators

Dodds

Sukochev

2007

Studia Math.

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“…For the special case that τ (1) < ∞ and E = L 1 , this result is proved in [Ran1] using somewhat different techniques.…”

Section: For the Measure Topology And There Exist Sequences {Pmentioning

confidence: 99%

Section: P-banach-saks Propertiesmentioning

confidence: 99%

Banach–Saks properties in symmetric spaces of measurable operators

Dodds

Sukochev

2007

Studia Math.

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“…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.…”

Section: Taking Limits Lim Infmentioning

confidence: 99%

Komlós’ theorem and the fixed point property for affine mappings

Benavides

Japón

2018

Proc. Amer. Math. Soc.

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Assume that X is a Banach space of measurable functions for which Komlós' Theorem holds. We associate to any closed convex bounded subset C of X a coefficient t(C) which attains its minimum value when C is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of t(C) ∈ [1, 2] and the value of the Lipschitz constants of the iterates. As a first consequence, for every L < 2, we deduce the existence of fixed points for affine uniformly L-Lipschitzian mappings defined on the closed unit ball of L1[0, 1]. Our main theorem also provides a wide collection of convex closed bounded sets in L 1 ([0, 1]) and in some other spaces of functions, which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in L1(µ) can only occur in the extremal case t(C) = 2. Examples are displayed proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient t(C).2010 Mathematics Subject Classification. 47H09, 47H10.

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L-embedded banach spaces and measure topology

Pfitzner

2014

Isr. J. Math.

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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].

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Non-commutative subsequence principles

Cited by 8 publications

References 26 publications

Banach–Saks properties in symmetric spaces of measurable operators

Banach–Saks properties in symmetric spaces of measurable operators

Komlós’ theorem and the fixed point property for affine mappings

L-embedded banach spaces and measure topology

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