{"title":"Lack of Metric Projectivity, Injectivity, and Flatness for Modules \\(L_p\\)","authors":"Norbert Nemesh","doi":"10.1134/S0016266324040038","DOIUrl":null,"url":null,"abstract":"<p> In this paper, we demonstrate that for a locally compact Hausdorff space <span>\\(S\\)</span> and a decomposable Borel measure <span>\\(\\mu\\)</span>, metric projectivity, injectivity, or flatness of the <span>\\(C_0(S)\\)</span>-module <span>\\(L_p(S,\\mu)\\)</span> implies that <span>\\(\\mu\\)</span> is purely atomic with at most one atom. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"58 4","pages":"371 - 383"},"PeriodicalIF":0.6000,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266324040038","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we demonstrate that for a locally compact Hausdorff space \(S\) and a decomposable Borel measure \(\mu\), metric projectivity, injectivity, or flatness of the \(C_0(S)\)-module \(L_p(S,\mu)\) implies that \(\mu\) is purely atomic with at most one atom.
期刊介绍：
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.
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