{"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":"<p>The Yosida representation for an Archimedean vector lattice <i>A</i> with weak unit <i>u</i>, denoted (<i>A</i>, <i>u</i>), reveals similarities between the ideas of the title, FST and SMP. If <i>A</i> is Archimedean, the conclusion of the FST means exactly that for each <span>\\(0 < e \\in A\\)</span>, the Yosida space for <span>\\((e^{dd},e)\\)</span>, denoted <span>\\(Y_e\\)</span>, has a base of clopen sets. This yields a short “Yosida based\" proof of FST. On the other hand, SMP implies that each <span>\\(Y_e\\)</span> has a <span>\\(\\pi \\)</span>-base of clopen sets. The converse fails, but holds if <i>A</i> has a strong unit (and in a somewhat more general situation).</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":\"<p>The Yosida representation for an Archimedean vector lattice <i>A</i> with weak unit <i>u</i>, denoted (<i>A</i>, <i>u</i>), reveals similarities between the ideas of the title, FST and SMP. If <i>A</i> is Archimedean, the conclusion of the FST means exactly that for each <span>\\\\(0 < e \\\\in A\\\\)</span>, the Yosida space for <span>\\\\((e^{dd},e)\\\\)</span>, denoted <span>\\\\(Y_e\\\\)</span>, has a base of clopen sets. This yields a short “Yosida based\\\" proof of FST. On the other hand, SMP implies that each <span>\\\\(Y_e\\\\)</span> has a <span>\\\\(\\\\pi \\\\)</span>-base of clopen sets. The converse fails, but holds if <i>A</i> has a strong unit (and in a somewhat more general situation).</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11117-024-01033-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11117-024-01033-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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)。
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).