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On Kalton’s interlaced graphs and nonlinear embeddings into dual Banach spaces
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-05-05, DOI: 10.1142/s1793525321500345
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 ([]k,d𝕂)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 ([]k,d𝕂)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 ([]k,d𝕂)k equi-Lipschitzly and so that Y has Szlenk index ω2. We prove that c0 does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than 32. We also show that neither c0 nor L1 coarsely embeds into a separable dual by a weak-to-weak sequentially continuous map.