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

Abstract

A recent result of T.~Abrahamsen, P.~Hájek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{\ast \ast \ast \ast }}$ such that $\Vert x+h\Vert =\max \{\Vert x\Vert ,1\}$ 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 $h\in S_{X^{\ast \ast \ast \ast }}$ 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 $h\in S_{X^{\ast \ast \ast \ast }}$ such that $\Vert x+h\Vert =\max \{\Vert x\Vert ,1\}$ for all $x\in X$.