Order denseness in free Banach lattices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912539575582720 |
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| author | Azouzi, Youssef Dhifaoui, Wassim |
| author_facet | Azouzi, Youssef Dhifaoui, Wassim |
| contents | We prove a fundamental property: the free vector lattice $FVL[E]$ over a Banach space E is order dense in the free p-convex Banach lattice $FBL^{(p)}[E],~~1 ^leq p \leq \infty,$ if and only if E is finite-dimensional. In a recent work, Oikhberg, Tradacete, Taylor, and Troitsky claimed that order denseness holds for all Banach spaces. We point out a gap in their proof, and consequently, any conclusions relying on this claim require reexamination -- a task we also undertake in this present paper. A key tool in our approach, which also leads to a partial answer to an open question recently posed by these authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11648 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Order denseness in free Banach lattices Azouzi, Youssef Dhifaoui, Wassim Functional Analysis 46B42 We prove a fundamental property: the free vector lattice $FVL[E]$ over a Banach space E is order dense in the free p-convex Banach lattice $FBL^{(p)}[E],~~1 ^leq p \leq \infty,$ if and only if E is finite-dimensional. In a recent work, Oikhberg, Tradacete, Taylor, and Troitsky claimed that order denseness holds for all Banach spaces. We point out a gap in their proof, and consequently, any conclusions relying on this claim require reexamination -- a task we also undertake in this present paper. A key tool in our approach, which also leads to a partial answer to an open question recently posed by these authors. |
| title | Order denseness in free Banach lattices |
| topic | Functional Analysis 46B42 |
| url | https://arxiv.org/abs/2508.11648 |