Free quasi-Banach lattices

TL;DR

This paper investigates free objects in quasi-Banach lattices, providing a functional representation for free p-convex p-Banach lattices and exploring their extension properties, structure, and density within free vector lattices.

Contribution

It introduces a functional representation for free p-convex p-Banach lattices generated by p-natural quasi-Banach spaces and analyzes their extension and structural properties.

Findings

Operators from Banach spaces extend to lattice homomorphisms with norm control

The space ℓ_p(Γ) is projective iff Γ is countable for 0<p<1

The free vector lattice embeds densely into the free p-convex p-Banach lattice

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.