弗赖登塔尔谱定理和阿基米德向量网格中的足够多投影

IF 0.8 3区数学 Q2 MATHEMATICS Positivity Pub Date : 2024-02-25 DOI:10.1007/s11117-024-01033-8

Anthony W. Hager, Brian Wynne

{"title":"弗赖登塔尔谱定理和阿基米德向量网格中的足够多投影","authors":"Anthony W. Hager, Brian Wynne","doi":"10.1007/s11117-024-01033-8","DOIUrl":null,"url":null,"abstract":"The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each \\(0 < e \\in A\\), the Yosida space for \\((e^{dd},e)\\), denoted \\(Y_e\\), has a base of clopen sets. This yields a short “Yosida based\" proof of FST. On the other hand, SMP implies that each \\(Y_e\\) has a \\(\\pi \\)-base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation).","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Freudenthal spectral theorem and sufficiently many projections in Archimedean vector lattices\",\"authors\":\"Anthony W. Hager, Brian Wynne\",\"doi\":\"10.1007/s11117-024-01033-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each \\\\(0 < e \\\\in A\\\\), the Yosida space for \\\\((e^{dd},e)\\\\), denoted \\\\(Y_e\\\\), has a base of clopen sets. This yields a short “Yosida based\\\" proof of FST. On the other hand, SMP implies that each \\\\(Y_e\\\\) has a \\\\(\\\\pi \\\\)-base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation).\",\"PeriodicalId\":54596,\"journal\":{\"name\":\"Positivity\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-02-25\",\"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/s11117-024-01033-8\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Positivity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11117-024-01033-8","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

摘要

具有弱单位 u 的阿基米德向量网格 A 的约西达表示（表示为 (A, u)）揭示了标题、FST 和 SMP 之间的相似性。如果 A 是阿基米德的，那么 FST 的结论就意味着，对于每个 \(0 < e \in A\), \((e^{dd},e)\) 的 Yosida 空间，表示为 \(Y_e\)，有一个开集的基。这就得到了一个简短的 "基于 Yosida "的 FST 证明。另一方面，SMP 意味着每个 \(Y_e\) 都有一个开集的基（\pi \）。反之亦然，但如果 A 有一个强单元则成立（在更一般的情况下）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

The Freudenthal spectral theorem and sufficiently many projections in Archimedean vector lattices

The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each \(0 < e \in A\), the Yosida space for \((e^{dd},e)\), denoted \(Y_e\), has a base of clopen sets. This yields a short “Yosida based" proof of FST. On the other hand, SMP implies that each \(Y_e\) has a \(\pi \)-base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation).

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Positivity 数学-数学

CiteScore

1.80

自引率

10.00%

发文量

审稿时长

>12 weeks

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.