Trond A. Abrahamsen, Ramón J. Aliaga, Vegard Lima, André Martiny, Yoël Perreau, Antonín Prochazka, Triinu Veeorg
{"title":"Delta-points and their implications for the geometry of Banach spaces","authors":"Trond A. Abrahamsen, Ramón J. Aliaga, Vegard Lima, André Martiny, Yoël Perreau, Antonín Prochazka, Triinu Veeorg","doi":"10.1112/jlms.12913","DOIUrl":null,"url":null,"abstract":"<p>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 <span></span><math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <mn>1</mn>\n </msub>\n <annotation>$\\ell _1$</annotation>\n </semantics></math>. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of <span></span><math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <mn>2</mn>\n </msub>\n <annotation>$\\ell _2$</annotation>\n </semantics></math>, with a <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space to have a <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-point. Next, we establish powerful relations between existence of <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points for the asymptotic geometry of Banach spaces. In addition, we prove that if <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is a Banach space with a shrinking <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math>-unconditional basis with <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo><</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$k &lt; 2$</annotation>\n </semantics></math>, or if <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> and its dual <span></span><math>\n <semantics>\n <msup>\n <mi>X</mi>\n <mo>∗</mo>\n </msup>\n <annotation>$X^*$</annotation>\n </semantics></math> fail to contain <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points. In particular, we get that no Lipschitz-free space with a Hahn–Banach smooth predual contains <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points. Finally, we present a purely metric characterization of the molecules in Lipschitz-free spaces that are <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points, and we solve an open problem about representation of finitely supported <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math>-points in Lipschitz-free spaces.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12913","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12913","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.