Bruno de Mendonça Braga, Gilles Lancien, Colin Petitjean, Antonín Procházka

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs ${({[\mathbb{N}]}^{k},{d}_{\mathbb{K}})}_{k}$ into dual spaces. Notably, we define and study a modification of Kalton’s property ${\mathcal{Q}}$ that we call property ${{\mathcal{Q}}}_{p}$ (with $p\in (1,+\infty ]$). We show that if ${({[\mathbb{N}]}^{k},{d}_{\mathbb{K}})}_{k}$ equi-coarse Lipschitzly embeds into ${X}^{\ast}$, then the Szlenk index of $X$ is greater than $\omega $, and that this is optimal, i.e. there exists a separable dual space ${Y}^{\ast}$ that contains ${({[\mathbb{N}]}^{k},{d}_{\mathbb{K}})}_{k}$ equi-Lipschitzly and so that $Y$ has Szlenk index ${\omega}^{2}$. We prove that ${c}_{0}$ does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than $\frac{3}{2}$. We also show that neither ${c}_{0}$ nor ${L}_{1}$ coarsely embeds into a separable dual by a weak-to-weak${}^{\ast}$ sequentially continuous map.