On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces

Abstract

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.