Free quasi-Banach lattices

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 ${\mathrm{FpBL}}^{(p)}[E]$ 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 $\mathrm{FBL}[E]\to X$ 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 ${\mathrm{FpBL}}^{(p)}[E]$ as a dense sublattice.

35 pages