Free quasi-Banach lattices
Alberto Salguero-Alarcón, Pedro Tradacete, Nazaret Trejo-Arroyo
TL;DR
The article develops a comprehensive theory of free objects in the realm of quasi-Banach spaces and lattices, centering on the free $p$-convex $p$-Banach lattice $ ext{FpBL}^{(p)}[E]$ for $p$-natural spaces $E$. It provides a functional representation into $L_p[0,1]$, proves extension properties for operators to lattice homomorphisms with norm control, and characterizes projectivity of $oldsymbol{ extell_p}(oldsymbol{ extGamma})$ in terms of the cardinality of $oldsymbol{ extGamma}$. The work also establishes the density of the free vector lattice inside the free convex object and develops a functorial perspective, while connecting to $L$-convex and non-$p$-convex free objects via Maurey–Nikishin factorization. These results illuminate how free-construction techniques extend beyond locally convex spaces and reveal structural links between convexity regimes, free objects, and projectivity in the quasi-Banach lattice setting. The paper closes with open questions and conjectures, notably regarding functional representations and chain conditions in broader contexts.
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 $\ell_p(Γ)$ $(0<p<1)$ 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.