Asymptotic smoothness in Banach spaces, three-space properties and applications

Causey, R.; Fovelle, A.; Lancien, G.

doi:10.1090/tran/8818

Asymptotic smoothness in Banach spaces, three-space properties and applications
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Trans. Amer. Math. Soc. 376 (2023), 1895-1928 Request permission

Abstract:

We study four asymptotic smoothness properties of Banach spaces, denoted $\mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p$, and $\mathsf {P}_p$. We complete their description by proving the missing renorming characterization for $\mathsf {A}_p$. We show that asymptotic uniform flattenability (property $\mathsf {T}_\infty$) and summable Szlenk index (property $\mathsf {A}_\infty$) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of $\mathsf {A}_p$ and $\mathsf {N}_p$ in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.

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