Asymptotic smoothness in Banach spaces, three space properties and applications

TL;DR

This paper investigates four asymptotic smoothness properties of Banach spaces, completes their characterization, and demonstrates that some are three space properties, providing new insights and rigidity results in Banach space theory.

Contribution

It completes the description of four asymptotic smoothness properties by proving a missing renorming theorem and establishes that two of these properties are three space properties.

Findings

Proves the missing renorming theorem for A_p.

Shows T_ ext{infty} and A_ ext{infty} are three space properties.

Derives new coarse Lipschitz rigidity results for A_p and N_p.

Abstract

We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and summable Szlenk index (property $\textsf{A}_\infty$) 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 $\textsf{A}_p$ and $\textsf{N}_p$ in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.