The free Banach lattices generated by $\ell_p$ and $c_0$

TL;DR

This paper characterizes the structure of free Banach lattices generated by _p and c_0, revealing their basis sequences form specific _r and _2 sequences, and explores their weakly compactly generated properties.

Contribution

It provides new insights into the structure of free Banach lattices generated by _p and c_0, including sequence estimates and compact generation properties.

Findings

Absolute values of basis form _r and _2 sequences

In any Banach lattice, _p sequences have upper _r estimates

Free Banach lattices generated by nonseparable _p() and c_0() are weakly compactly generated

Abstract

We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the absolute values of any $\ell_p$ sequence always have an upper $\ell_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $\ell_p(\Gamma)$ for $2<p<\infty$, as well as $c_0(\Gamma)$, are weakly compactly generated whereas this is not the case for $1\leq p\leq 2$.