{"title":"Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1","authors":"E. Casini, E. Miglierina, Łukasz Piasecki","doi":"10.12775/tmna.2023.009","DOIUrl":null,"url":null,"abstract":"We provide a concrete isometric description of all the preduals of $\\ell_1$ \nfor which the standard basis in $\\ell_1$ has a finite number of $w^*$-limit points.\n Then, we apply this result to give an example of an $\\ell_1$-predual $X$ such\n that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive\n mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy \nof any hyperplane $W_{\\alpha}$ of the space $c$ of convergent sequences such\n that $W_\\alpha$ is a predual of $\\ell_1$ and $W_\\alpha^*$ lacks the $w^*$-FPP.\n This answers a question left open in the 2017 paper of the present authors.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topological Methods in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.12775/tmna.2023.009","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We provide a concrete isometric description of all the preduals of $\ell_1$
for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points.
Then, we apply this result to give an example of an $\ell_1$-predual $X$ such
that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive
mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy
of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such
that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP.
This answers a question left open in the 2017 paper of the present authors.
期刊介绍:
Topological Methods in Nonlinear Analysis (TMNA) publishes research and survey papers on a wide range of nonlinear analysis, giving preference to those that employ topological methods. Papers in topology that are of interest in the treatment of nonlinear problems may also be included.