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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 $([ℕ]k,d𝕂)k$ 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 $([ℕ]k,d𝕂)k$ 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 $([ℕ]k,d𝕂)k$ 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.