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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.02851 |
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| _version_ | 1866911348669022208 |
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| author | Zabeti, Omid |
| author_facet | Zabeti, Omid |
| contents | For an arbitrary topological space $X$, assume that $S(X)$ is the vector lattice of all equivalence classes of real-valued continuous functions on open dense subsets of $X$; it is a laterally complete vector lattice but not a normed lattice, certainly. Nevertheless, we can have the extended unbounded norm topology ($un$-topology) on it. On the other hand, by a remarkable result of Wickstead, there exists a representation approach for every Archimedean vector lattice $E$ in terms of $S(X)$-spaces. In this paper, we show that this representation is order continuous and when $E$ is order complete, it coincides with the known Maeda-Ogasawara representation. Moreover, when $E$ is a Banach lattice, by consideration of the $un$-topology on $E$ and the extended $un$-topology on $S(X)$, we show that this representation is, in fact, a homeomorphism. With the aid of this topological attitude, we establish a representation theorem (in fact a homeomorphism) for the Fremlin projective tensor product between Banach lattices, in terms of $S(X)$-spaces, as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Order Continuous and Topological Representations of Archimedean Vector Lattices via $S(X)$-spaces Zabeti, Omid Functional Analysis For an arbitrary topological space $X$, assume that $S(X)$ is the vector lattice of all equivalence classes of real-valued continuous functions on open dense subsets of $X$; it is a laterally complete vector lattice but not a normed lattice, certainly. Nevertheless, we can have the extended unbounded norm topology ($un$-topology) on it. On the other hand, by a remarkable result of Wickstead, there exists a representation approach for every Archimedean vector lattice $E$ in terms of $S(X)$-spaces. In this paper, we show that this representation is order continuous and when $E$ is order complete, it coincides with the known Maeda-Ogasawara representation. Moreover, when $E$ is a Banach lattice, by consideration of the $un$-topology on $E$ and the extended $un$-topology on $S(X)$, we show that this representation is, in fact, a homeomorphism. With the aid of this topological attitude, we establish a representation theorem (in fact a homeomorphism) for the Fremlin projective tensor product between Banach lattices, in terms of $S(X)$-spaces, as well. |
| title | Order Continuous and Topological Representations of Archimedean Vector Lattices via $S(X)$-spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2404.02851 |