Free quasi-Banach lattices
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911303738589184 |
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| author | Salguero-Alarcón, Alberto Tradacete, Pedro Trejo-Arroyo, Nazaret |
| author_facet | Salguero-Alarcón, Alberto Tradacete, Pedro Trejo-Arroyo, Nazaret |
| contents | We study different versions of \emph{free objects} in the setting of quasi-Banach spaces and quasi-Banach lattices. Special attention is devoted to the free $p$-convex $p$-Banach lattice $\operatorname{FpBL}^{(p)}[E]$ generated by a $p$-natural quasi-Banach space $E$, for which we provide a functional representation by means of operators into $L_p[0,1]$. This representation yields, among other consequences: (1) Operators from a Banach space $E$ to any $p$-convex $(0<p<1)$ quasi-Banach lattice $X$ can be extended to lattice homomorphisms $\operatorname{FBL}[E] \to X$ with control of the norm. (2) The space $\ell_p(Γ)$ $(0<p<1)$ is a projective $p$-Banach lattice precisely when $Γ$ is countable. (3) The free vector lattice generated by $E$ sits inside $\operatorname{FpBL}^{(p)}[E]$ as a dense sublattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05273 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free quasi-Banach lattices Salguero-Alarcón, Alberto Tradacete, Pedro Trejo-Arroyo, Nazaret Functional Analysis Primary: 46A40, 06B25, 46B42. Secondary: 46M10, 47B60 We study different versions of \emph{free objects} in the setting of quasi-Banach spaces and quasi-Banach lattices. Special attention is devoted to the free $p$-convex $p$-Banach lattice $\operatorname{FpBL}^{(p)}[E]$ generated by a $p$-natural quasi-Banach space $E$, for which we provide a functional representation by means of operators into $L_p[0,1]$. This representation yields, among other consequences: (1) Operators from a Banach space $E$ to any $p$-convex $(0<p<1)$ quasi-Banach lattice $X$ can be extended to lattice homomorphisms $\operatorname{FBL}[E] \to X$ with control of the norm. (2) The space $\ell_p(Γ)$ $(0<p<1)$ is a projective $p$-Banach lattice precisely when $Γ$ is countable. (3) The free vector lattice generated by $E$ sits inside $\operatorname{FpBL}^{(p)}[E]$ as a dense sublattice. |
| title | Free quasi-Banach lattices |
| topic | Functional Analysis Primary: 46A40, 06B25, 46B42. Secondary: 46M10, 47B60 |
| url | https://arxiv.org/abs/2512.05273 |