Delta-points and their implications for the geometry of Banach spaces
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arXiv
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| Format: | Preprint |
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2023
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| author | Abrahamsen, Trond A. Aliaga, Ramón J. Lima, Vegard Martiny, André Perreau, Yoël Prochazka, Antonín Veeorg, Triinu |
| author_facet | Abrahamsen, Trond A. Aliaga, Ramón J. Lima, Vegard Martiny, André Perreau, Yoël Prochazka, Antonín Veeorg, Triinu |
| contents | 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 $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point.
Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points.
Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00511 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Delta-points and their implications for the geometry of Banach spaces Abrahamsen, Trond A. Aliaga, Ramón J. Lima, Vegard Martiny, André Perreau, Yoël Prochazka, Antonín Veeorg, Triinu Functional Analysis 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 $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point. Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces. |
| title | Delta-points and their implications for the geometry of Banach spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2303.00511 |