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 [Formula: see text] into dual spaces. Notably, we define and study a modification of Kalton’s property [Formula: see text] that we call property [Formula: see text] (with [Formula: see text]). We show that if [Formula: see text] equi-coarse Lipschitzly embeds into [Formula: see text], then the Szlenk index of [Formula: see text] is greater than [Formula: see text], and that this is optimal, i.e. there exists a separable dual space [Formula: see text] that contains [Formula: see text] equi-Lipschitzly and so that [Formula: see text] has Szlenk index [Formula: see text]. We prove that [Formula: see text] does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than [Formula: see text]. We also show that neither [Formula: see text] nor [Formula: see text] coarsely embeds into a separable dual by a weak-to-weak[Formula: see text] sequentially continuous map.