{"title":"On maximal solid subspaces of intermediate algebras in C(X)","authors":"J. M. Domínguez","doi":"10.1007/s11117-024-01067-y","DOIUrl":null,"url":null,"abstract":"<p>Let <i>C</i>(<i>X</i>) be the algebra of all real-valued continuous functions on a Tychonoff space <i>X</i>, and <span>\\(C^*(X)\\)</span> the subalgebra of bounded functions. We prove that if <i>B</i> is any subalgebra of <i>C</i>(<i>X</i>) containing <span>\\(C^*(X)\\)</span>, then no maximal solid subspace of <i>B</i> contains <span>\\(C^*(X)\\)</span>, and we derive from this that the maximal solid subspaces of <i>B</i> are exactly the real maximal ideals of <i>B</i>. Then we extend the above to the case of intermediate algebras in <i>A</i>, where <i>A</i> is a <span>\\(\\varPhi \\)</span>-algebra with bounded inversion.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Positivity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11117-024-01067-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let C(X) be the algebra of all real-valued continuous functions on a Tychonoff space X, and \(C^*(X)\) the subalgebra of bounded functions. We prove that if B is any subalgebra of C(X) containing \(C^*(X)\), then no maximal solid subspace of B contains \(C^*(X)\), and we derive from this that the maximal solid subspaces of B are exactly the real maximal ideals of B. Then we extend the above to the case of intermediate algebras in A, where A is a \(\varPhi \)-algebra with bounded inversion.
让 C(X) 是泰克诺夫空间 X 上所有实值连续函数的代数，\(C^*(X)\) 是有界函数的子代数。我们证明，如果 B 是 C(X) 的任何包含 \(C^*(X)\) 的子代数，那么 B 的最大实体子空间都不包含 \(C^*(X)\)，我们由此推导出 B 的最大实体子空间正是 B 的实最大ideals。然后，我们把上面的方法推广到 A 中的中间代数的情况，其中 A 是一个有界反转的 \(\varPhi \)-代数。
期刊介绍：
The purpose of Positivity is to provide an outlet for high quality original research in all areas of analysis and its applications to other disciplines having a clear and substantive link to the general theme of positivity. Specifically, articles that illustrate applications of positivity to other disciplines - including but not limited to - economics, engineering, life sciences, physics and statistical decision theory are welcome.
The scope of Positivity is to publish original papers in all areas of mathematics and its applications that are influenced by positivity concepts.
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