弗赖登塔尔谱定理和阿基米德向量网格中的足够多投影

Pub Date : 2024-02-25 DOI:10.1007/s11117-024-01033-8

Anthony W. Hager, Brian Wynne

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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）。

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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).

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