Banach spaces containing \(c_0\) and elements in the fourth dual (Q2069937)

A Banach space \(X\) with unit sphere \(S_X\) is said to be almost square (ASQ) if for every \(\varepsilon > 0\) and every \(x_1, x_2, \ldots, x_n \in S_X,\) there exists \(h \in S_X\) with \(\|x_i \pm h\| \le 1 + \varepsilon\) for every \(i = 1, 2, \ldots, n.\) The ASQ property has been studied in several papers, and recently in [\textit{T. A. Abrahamsen} et al., J. Math. Anal. Appl. 487, No. 2, Article ID 124003, 11 p. (2020; Zbl 1442.46005)] a characterization of this property involving the fourth dual was obtained. This characterization reads: A separable (real) Banach space \(X\) is ASQ if and only if there exists \(h \in S_{X^{****}}\) such that \(\|x + h\| = \max\{\|x\|, 1\}\) for all \(x \in X.\) The two main theorems in the paper under review extend the aforementioned result both in the isomorphic and isometric direction in the following way: \begin{itemize} \item Let \(X\) be a Banach space that contains \(c_0.\) Then there exists an equivalent norm \(|\cdot|\) on \(X\) for which there exists \(h \in S_{X^{****}}\) such that \(|x + h| = \max\{|x|, 1\}\) for all \(x \in X\) (Theorem~1.4). \item Let \(X\) be a sequentially ASQ Banach space and let \(\mathcal U\) be a selective ultrafilter on the natural numbers. Then there exists \(h \in S_{X^{****}}\) such that \(\|x + h\| = \max\{\|x\|, 1\}\) for all \(x \in X\) (Theorem~1.5). \end{itemize} It is known from [\textit{J. Becerra Guerrero} et al., J. Math. Anal. Appl. 438, No. 2, 1030--1040 (2016; Zbl 1346.46005)] that a Banach space contains \(c_0\) if and only if it admits an equivalent ASQ norm. Moreover, a Banach space \(X\) is sequentially ASQ (a property introduced in the paper under review) if there exists a sequence \((x_n)\) in \(S_X\) such that for every \(x \in X\) we have \(\lim_n\|x + x_n\| = \max\{\|x\|,1\}.\) Every sequentially ASQ space is ASQ and the converse is true for separable spaces, but not true in general (see Proposition~2.3 and below). As for selective ultrafilters on the natural numbers, it is known that their existence is independent of ZFC and that they exist under CH (but also under less restrictive assumptions).