{"title":"Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on","authors":"Shirin Tavkoli, Rasoul Abazari, Ali Jabbari","doi":"10.1155/2024/5893357","DOIUrl":null,"url":null,"abstract":"Let <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-89\"></use></g></svg> be a nonempty set, <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.2729 8.68572\" width=\"9.2729pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-66\"></use></g></svg> be a commutative Banach algebra, and <span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.34882 17.503 11.7782\" width=\"17.503pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,9.872,0)\"></path></g></svg><span></span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"21.085183800000003 -8.34882 18.973 11.7782\" width=\"18.973pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.135,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.477,0)\"></path></g></svg><span></span><span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"43.6901838 -8.34882 13.517 11.7782\" width=\"13.517pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,43.74,0)\"></path></g></svg>.</span></span> In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of <span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.541 28.884 12.8091\" width=\"28.884pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-127\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.047,-5.741)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.998,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.496,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.92,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"31.0131838 -10.541 13.873 12.8091\" width=\"13.873pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,31.063,0)\"><use xlink:href=\"#g113-66\"></use></g><g transform=\"matrix(.013,0,0,-0.013,40.198,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>.</span></span> Specifically, we establish that <span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.541 28.884 12.8091\" width=\"28.884pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-127\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.047,-5.741)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.998,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.496,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.92,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"31.0131838 -10.541 13.873 12.8091\" width=\"13.873pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,31.063,0)\"><use xlink:href=\"#g113-66\"></use></g><g transform=\"matrix(.013,0,0,-0.013,40.198,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> possesses the BSE property if and only if <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-89\"></use></g></svg> is finite and <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.2729 8.68572\" width=\"9.2729pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-66\"></use></g></svg> is BSE. Additionally, we investigate the BSE module property on Banach <span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.541 28.884 12.8091\" width=\"28.884pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-127\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.047,-5.741)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.998,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.496,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.92,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"31.0131838 -10.541 13.873 12.8091\" width=\"13.873pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,31.063,0)\"><use xlink:href=\"#g113-66\"></use></g><g transform=\"matrix(.013,0,0,-0.013,40.198,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>modules and demonstrate that a Banach space <span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.541 28.884 12.8091\" width=\"28.884pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-127\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.047,-5.741)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.998,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.496,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.92,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"31.0131838 -10.541 13.208 12.8091\" width=\"13.208pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,31.063,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,39.544,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> serves as a BSE Banach <span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"-0.0498162 -10.541 28.884 12.8091\" width=\"28.884pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-127\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.047,-5.741)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.998,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.496,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.92,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"12.8091pt\" style=\"vertical-align:-2.2681pt\" version=\"1.1\" viewbox=\"31.0131838 -10.541 13.873 12.8091\" width=\"13.873pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,31.063,0)\"><use xlink:href=\"#g113-66\"></use></g><g transform=\"matrix(.013,0,0,-0.013,40.198,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>module if and only if <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-89\"></use></g></svg> is finite and <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.6074 8.68572\" width=\"8.6074pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-90\"></use></g></svg> represents a BSE Banach <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.2729 8.68572\" width=\"9.2729pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-66\"></use></g></svg>-</span>module.","PeriodicalId":15840,"journal":{"name":"Journal of Function Spaces","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Function Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/5893357","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
Let be a nonempty set, be a commutative Banach algebra, and . In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of . Specifically, we establish that possesses the BSE property if and only if is finite and is BSE. Additionally, we investigate the BSE module property on Banach -modules and demonstrate that a Banach space serves as a BSE Banach -module if and only if is finite and represents a BSE Banach -module.
期刊介绍：
Journal of Function Spaces (formerly titled Journal of Function Spaces and Applications) publishes papers on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. As well as original research, Journal of Function Spaces also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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