Big slices versus big relatively weakly open subsets in Banach spaces

Guerrero, Julio Becerra; López-Pérez, Ginés; Zoca, Abraham Rueda

doi:10.1016/j.jmaa.2015.03.056

Cited by 28 publications

(28 citation statements)

References 14 publications

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“…The answer is no, in fact, every Daugavet-point in D B has a weak neighborhood of arbitrary small diameter. Let us remark that the first example of a Banach space with the local diameter two property, but failing the diameter two property was given in [BGLPRZ15]. While we have used binary trees, their construction used the tree of finite sequences of positive integers and they even showed that every Banach space containing c 0 can be renormed to have the local diameter two property and fail the diameter two property.…”

Section: A Space With 1-unconditional Basis and Daugavet-pointsmentioning

confidence: 99%

Daugavet- and delta-points in Banach spaces with unconditional bases

Abrahamsen¹,

Lima²,

Martiny³

et al. 2021

Trans. Amer. Math. Soc. Ser. B

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We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1 1 -unconditional basis. A norm one element x x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 2 from x x . A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2 2 . We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1 1 -unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a 1 1 -unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

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Section: A Space With 1-unconditional Basis and Daugavet-pointsmentioning

confidence: 99%

Daugavet- and delta-points in Banach spaces with unconditional bases

Abrahamsen¹,

Lima²,

Martiny³

et al. 2021

Trans. Amer. Math. Soc. Ser. B

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“…Thus the SD2P implies that every non-empty relatively weakly open subset of B X has diameter 2, which in turn implies that every slice of B X has diameter 2. None of these implications is reversible ( [5], [12]).…”

Section: Introductionmentioning

confidence: 99%

New applications of extremely regular function spaces

2019

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Let L be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of C0(L) have very strong diameter 2 properties and, for every real number ε with 0 < ε < 1, contain an ε-isometric copy of c0. If L does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of ℓ1.2010 Mathematics Subject Classification. 46B20; 46B22.

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“…These properties, which are extremely opposite to the Radon-Nikodým property, have been deeply studied over the last few years. For instance, it was recently proved [3,4] that each one of the above properties is different from the rest in an extreme way.…”

Section: Introductionmentioning

confidence: 99%

Octahedrality in Lipschitz-free Banach spaces

Guerrero¹,

López-Pérez²,

Zoca³

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it by analysing the weak-star strong diameter two property in Lipschitz functions spaces. Also, we show an example which proves that our results are optimal and that octahedrality in vectorvalued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B22, 52A10.

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Big slices versus big relatively weakly open subsets in Banach spaces

Cited by 28 publications

References 14 publications

Daugavet- and delta-points in Banach spaces with unconditional bases

Daugavet- and delta-points in Banach spaces with unconditional bases

New applications of extremely regular function spaces

Octahedrality in Lipschitz-free Banach spaces

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