Delta-points in Banach spaces generated by adequate families

Abrahamsen, Trond A.; Lima, Vegard; Martiny, André

doi:10.1215/00192082-10123638

Cited by 2 publications

(2 citation statements)

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“…The proof of this theorem relies on Theorem 5.6 and the theorem is used to show that if a Banach space 𝑋 has a shrinking 𝑘-unconditional basis for 𝑘 < 2, then 𝑋 contains no 𝔇-points and 𝑋 * contains no weak * Δ-points (see Corollary 5.15). This result answers [6,Question 5.6] affirmatively and can be used to strengthen [6,Proposition 4.6]. If 𝑋 has a monotone boundedly complete 𝑘-unconditional basis for 𝑘 < 2, we only get that 𝑋 has no Δpoints.…”

Section: Introductionsupporting

confidence: 65%

“…Among the first nonreflexive examples of Banach spaces with no Δ-points were some Membedded spaces and their duals, for example, 𝑐 0 and 𝓁 1 , (𝓁 2 ) and its dual, and the Schreier space and its dual. For the Schreier space and its dual, this was shown in [6], but it also follows from Corollary 5.15. The first two are covered by Corollary 5.9 and, in fact, both 𝑐 0 and (𝓁 2 ) have Kalton's property (M * ).…”

Section: Sequential Super Points and Hahn-banach Smooth Spacesmentioning

confidence: 71%

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Delta‐points and their implications for the geometry of Banach spaces

Abrahamsen,

Aliaga,

Lima

et al. 2024

Journal of London Math Soc

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We show that the Lipschitz‐free space with the Radon–Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to . Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of , with a ‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a ‐point. Next, we establish powerful relations between existence of ‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of ‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if is a Banach space with a shrinking ‐unconditional basis with , or if is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then and its dual fail to contain ‐points. In particular, we get that no Lipschitz‐free space with a Hahn–Banach smooth predual contains ‐points. Finally, we present a purely metric characterization of the molecules in Lipschitz‐free spaces that are ‐points, and we solve an open problem about representation of finitely supported ‐points in Lipschitz‐free spaces.

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Section: Introductionsupporting

confidence: 65%

Section: Sequential Super Points and Hahn-banach Smooth Spacesmentioning

confidence: 71%