Banach spaces containing $c_0$ and elements in the fourth dual

TL;DR

This paper investigates the structure of nonseparable Banach spaces containing $c_0$, exploring conditions under which certain dual space elements characterize these spaces, and extends results from separable to nonseparable contexts using ultrafilters.

Contribution

It establishes a characterization of Banach spaces containing $c_0$ via renormings and dual elements, and extends the concept of almost square spaces to the nonseparable setting under certain set-theoretic assumptions.

Findings

A Banach space contains $c_0$ iff it admits a renorming with a specific dual element property.

Under the existence of selective ultrafilters, sequentially almost square spaces have a dual element characterization.

The nonseparable extension of the almost square characterization remains open without additional set-theoretic assumptions.

Abstract

A recent result of T.~Abrahamsen, P.~H\'ajek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes through a sequential version of being almost square which we call being \textit{sequentially almost square}. In this article we study these conditions in the nonseparable setting. On one hand, we show that a Banach space $X$ contains a copy of $c_0$ if and only if there exists an equivalent renorming $| \cdot |$ on $X$ for which there exists $h\in S_{X^{****}}$ such that $|x+h|=\max\{|x|,1\}$ for every $x\in X$. On the other hand, although it is unclear whether the aforementioned result of T.~Abrahamsen et al. holds in the nonseparable setting, we show that, under the existence of selective ultrafilters, if $X$ is a sequentially almost square Banach space then there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$.