{"title":"On the Schauder fixed point property II","authors":"Khadime Salame","doi":"10.1007/s43034-023-00300-1","DOIUrl":null,"url":null,"abstract":"<div><p>The Schauder fixed point property (<b>F</b>) was introduced and studied by Lau and Zhang as a semigroup formulation in the general setting of convex spaces of the well-known Schauder fixed point theorem in Banach spaces. What amenability property should possess a semigroup or a topological group to satisfy the Schauder fixed point property. Recently, the author provided a partial answer to that question and as a sequel, it is the purpose of this paper to study in more deep this problem. Our main result establishes that for a compact semitopological semigroup <i>S</i> we have: LUC(<i>S</i>) is left amenable if, and only if, <i>S</i> has the fixed point property (<b>F</b>). Furthermore, we also prove that totally bounded topological groups, semitopological groups <i>S</i> with the property that LUC(<i>S</i>) <span>\\(\\subset \\)</span><span>\\({\\textrm{aa}}\\)</span>(<i>S</i>), and strongly left amenable semitopological semigroups, possess all the Schauder fixed point property.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00300-1","RegionNum":3,"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
The Schauder fixed point property (F) was introduced and studied by Lau and Zhang as a semigroup formulation in the general setting of convex spaces of the well-known Schauder fixed point theorem in Banach spaces. What amenability property should possess a semigroup or a topological group to satisfy the Schauder fixed point property. Recently, the author provided a partial answer to that question and as a sequel, it is the purpose of this paper to study in more deep this problem. Our main result establishes that for a compact semitopological semigroup S we have: LUC(S) is left amenable if, and only if, S has the fixed point property (F). Furthermore, we also prove that totally bounded topological groups, semitopological groups S with the property that LUC(S) \(\subset \)\({\textrm{aa}}\)(S), and strongly left amenable semitopological semigroups, possess all the Schauder fixed point property.
期刊介绍:
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory.
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