Asymptotic Unconditionality

TL;DR

This paper characterizes when a separable real Banach space can be embedded almost isometrically into a space with a shrinking 1-unconditional basis, linking geometric properties to weak*-null sequences.

Contribution

It provides a new characterization of embeddings into spaces with shrinking 1-unconditional bases using asymptotic unconditionality conditions.

Findings

Characterization of embeddings via limits involving weak*-null sequences

Extension to reflexive spaces with reflexive embeddings

Connection to recent work of Johnson and Zheng

Abstract

We show that a separable real Banach space embeds almost isometrically in a space $Y$ with a shrinking 1-unconditional basis if and only if $\lim_{n \to \infty} \|x^* + x_n^*\| = \lim_{n \to \infty} \|x^* - x_n^*\|$ whenever $x^* \in X^*$, $(x_n^*)$ is a weak$^*$-null sequence and both limits exist. If $X$ is reflexive then $Y$ can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.