Sufficiently many projections in archimedean vector lattices with weak order unit
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914773455601664 |
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| author | Hager, Anthony W. Wynne, Brian |
| author_facet | Hager, Anthony W. Wynne, Brian |
| contents | The property of a vector lattice of sufficiently many projections (SMP) is informed by restricting attention to archimedean $A$ with a distinguished weak order unit $u$ (the class, or category, $\bf{W}$), where the Yosida representation $A \leq D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to $Y(A,u)$ having a $π$-base of clopen sets of a certain type called ``local". If the unit is strong, all clopen sets are local and $A$ is SMP if and only if $Y(A,u)$ has clopen $π$-base, a property we call $π$-zero-dimensional ($π$ZD). The paper is in two parts: the first explicates the similarities of SMP and $π$ZD; the second consists of examples, including $π$ZD but not SMP, and constructions of many SMP's which seem scarce in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sufficiently many projections in archimedean vector lattices with weak order unit Hager, Anthony W. Wynne, Brian General Topology 46A40, 06F20, 54D35, 54H10 The property of a vector lattice of sufficiently many projections (SMP) is informed by restricting attention to archimedean $A$ with a distinguished weak order unit $u$ (the class, or category, $\bf{W}$), where the Yosida representation $A \leq D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to $Y(A,u)$ having a $π$-base of clopen sets of a certain type called ``local". If the unit is strong, all clopen sets are local and $A$ is SMP if and only if $Y(A,u)$ has clopen $π$-base, a property we call $π$-zero-dimensional ($π$ZD). The paper is in two parts: the first explicates the similarities of SMP and $π$ZD; the second consists of examples, including $π$ZD but not SMP, and constructions of many SMP's which seem scarce in the literature. |
| title | Sufficiently many projections in archimedean vector lattices with weak order unit |
| topic | General Topology 46A40, 06F20, 54D35, 54H10 |
| url | https://arxiv.org/abs/2404.17628 |