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

TL;DR

This paper investigates the nonlinear embeddability of Kalton's interlaced graphs into dual Banach spaces, establishing new conditions related to the Szlenk index and demonstrating limitations on embeddings of classical spaces like c0 and L1.

Contribution

It introduces the property alQ_p, links graph embeddability to Szlenk index bounds, and shows non-embeddability results for c0 and L1 into dual spaces.

Findings

Equi-coarse Lipschitz embedding implies Szlenk index > 

Existence of dual spaces with Szlenk index ^2 containing the graphs

c0 does not embed with distortion < 3/2 into separable duals

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 $X^*$, then the Szlenk index of $X$ is greater than $\omega$, and that this is optimal, i.e., there exists a separable dual space $Y^*$ 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$^*$ sequentially continuous map.