This is an announcement for the paper “The free Banach lattices generated by $\ell_p$ and $c_0$” by Antonio Aviléshttps://arxiv.org/search/math?searchtype=author&query=Avil%C3%A9s%2C+A, Pedro Tradacetehttps://arxiv.org/search/math?searchtype=author&query=Tradacete%2C+P, Ignacio Villanuevahttps://arxiv.org/search/math?searchtype=author&query=Villanueva%2C+I.
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 $1\leq p\leq 2$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.02553