Daugavet- and Delta-points in absolute sums of Banach spaces

Haller, Rainis; Pirk, Katriin; Veeorg, Triinu

doi:10.48550/arxiv.2001.06197

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…The results of [2, Section 3] show that, in many classical Banach spaces, the concept of ∆-and Daugavet point coincide. The first example of a ∆-point which is not a Daugavet point [2, Example 4.7] required a study of absolute normalised norms (which was pushed quite further in [15]). See also [3] for more technical examples of Banach spaces containing ∆-points which are not Daugavet points.…”

Section: ∆-Pointsmentioning

confidence: 99%

Daugavet points and $Δ$-points in Lipschitz-free spaces

Jung¹,

Zoca²

2020

Preprint

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We study Daugavet points and ∆-points in Lipschitz-free Banach spaces. We prove that, if M is a compact metric space, then µ ∈ S F (M ) is a Daugavet point if, and only if, there is no denting point of B F (M ) at distance strictly smaller than two from µ. Moreover, we prove that if x and y are connectable by rectifiable curves of lenght as close to d(x, y) as we wish, then the molecule m x,y is a ∆-point. Some conditions on M which guarantee that the previous implication reverses are also obtained. As a consequence of our work, we show that Lipschitz-free spaces are natural examples of Banach spaces where we can guarantee the existence of ∆-points which are not Daugavet points.

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Section: ∆-Pointsmentioning

confidence: 99%

Daugavet points and $Δ$-points in Lipschitz-free spaces

Jung¹,

Zoca²

2020

Preprint

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“…It is immediate that every Daugavet-point is a ∆-point but, in general, a ∆-point does not need to be a Daugavet-point [1,Example 4.7]. See [1,19] for background and motivation for the study of Daugavet-points and ∆-points.…”

Section: Definition 31 ([1]mentioning

confidence: 99%

Daugavet property in projective symmetric tensor products of Banach spaces

Martín¹,

Zoca²

2020

Preprint

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We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L 1 -predual (i.e. X * is isometric to an L 1 -space) or X is a vector-valued L 1 -space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e. Daugavet points and ∆-points introduced in [1]) for L 1 -preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L 1preduals in terms of the absence of ∆-points and also to provide new examples of L 1 -preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces (in the sense of [40]) so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L 1 -preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in [34] about the Daugavet property for projective tensor products is also got.

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Daugavet- and Delta-points in absolute sums of Banach spaces

Cited by 2 publications

References 6 publications

Daugavet points and $Δ$-points in Lipschitz-free spaces

Daugavet points and $Δ$-points in Lipschitz-free spaces

Daugavet property in projective symmetric tensor products of Banach spaces

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