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arXiv · 2512.05273

Free quasi-Banach lattices

Alberto Salguero-Alarcón·Pedro Tradacete·Nazaret Trejo-Arroyo
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Abstract

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 $\ell_p(\Gamma)$ $(0<p<1)$ is a projective $p$-Banach lattice precisely when $\Gamma$ is countable. (3) The free vector lattice generated by $E$ sits inside $\operatorname{FpBL}^{(p)}[E]$ as a dense sublattice.

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Alberto Salguero-Alarcón, Pedro Tradacete, Nazaret Trejo-Arroyo. 2025-12-04. Free quasi-Banach lattices. https://arxiv.org/abs/2512.05273

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