{"title":"The BSE property for some vector-valued Banach function algebras","authors":"Fatemeh Abtahi, Ali Rejali, Farshad Sayaf","doi":"10.1007/s10473-024-0518-z","DOIUrl":null,"url":null,"abstract":"<p>In this paper, <i>X</i> is a locally compact Hausdorff space and <span>\\({\\cal A}\\)</span> is a Banach algebra. First, we study some basic features of <i>C</i><sub>0</sub>(<i>X</i>, <span>\\({\\cal A}\\)</span>) related to BSE concept, which are gotten from <span>\\({\\cal A}\\)</span>. In particular, we prove that if <i>C</i><sub>0</sub>(<i>X</i>, <span>\\({\\cal A}\\)</span>) has the BSE property then <span>\\({\\cal A}\\)</span> has so. We also establish the converse of this result, whenever <i>X</i> is discrete and <span>\\({\\cal A}\\)</span> has the BSE-norm property. Furthermore, we prove the same result for the BSE property of type I. Finally, we prove that <i>C</i><sub>0</sub> (<i>X</i>, <span>\\({\\cal A}\\)</span>) has the BSE-norm property if and only if <span>\\({\\cal A}\\)</span> has so.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"4 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0518-z","RegionNum":4,"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
In this paper, X is a locally compact Hausdorff space and \({\cal A}\) is a Banach algebra. First, we study some basic features of C0(X, \({\cal A}\)) related to BSE concept, which are gotten from \({\cal A}\). In particular, we prove that if C0(X, \({\cal A}\)) has the BSE property then \({\cal A}\) has so. We also establish the converse of this result, whenever X is discrete and \({\cal A}\) has the BSE-norm property. Furthermore, we prove the same result for the BSE property of type I. Finally, we prove that C0 (X, \({\cal A}\)) has the BSE-norm property if and only if \({\cal A}\) has so.
期刊介绍:
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.