On Kalton’s interlaced graphs and nonlinear embeddings into dual Banach spaces
Résumé
We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton's interlaced graphs ([N] k , d K) k into dual spaces. Notably, we define and study a modification of Kalton's property Q that we call property Qp (with p ∈ (1, +∞]). We show that if ([N] k , d K) k equi-coarse Lipschitzly embeds into X * , then the Szlenk index of X is greater than ω, and that this is optimal, i.e., there exists a separable dual space Y * that contains ([N] k , d K) k equi-Lipschitzly and so that Y has Szlenk index ω 2. We prove that c0 does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than 3 2. We also show that neither c0 nor L1 coarsely embeds into a separable dual by a weak-to-weak * sequentially continuous map.
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