{"title":"Band projections and order idempotents in Banach lattice algebras","authors":"David Muñoz-Lahoz","doi":"10.1016/j.jmaa.2024.129045","DOIUrl":null,"url":null,"abstract":"<div><div>Motivated by recent work about band projections on spaces of regular operators over a Banach lattice, given a Banach lattice algebra <em>A</em>, we will say an element <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> is a band projection if the multiplication operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a band projection. Our aim in this note is to explore the relations between this and the notion of order idempotent (those elements <em>a</em> in a Banach lattice algebra <em>A</em> with identity <em>e</em> such that <span><math><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>e</mi></math></span> and <span><math><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>a</mi></math></span>). We also revisit the properties of the ideal generated by the identity on a Banach lattice algebra, motivated by those of the centre of a Banach lattice.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129045"},"PeriodicalIF":1.2000,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009673","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
Motivated by recent work about band projections on spaces of regular operators over a Banach lattice, given a Banach lattice algebra A, we will say an element is a band projection if the multiplication operator is a band projection. Our aim in this note is to explore the relations between this and the notion of order idempotent (those elements a in a Banach lattice algebra A with identity e such that and ). We also revisit the properties of the ideal generated by the identity on a Banach lattice algebra, motivated by those of the centre of a Banach lattice.
期刊介绍:
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