Banach spaces containing c0 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∈SX⁎⁎⁎⁎ such that ‖x+h‖=max⁡{‖x‖,1} for all x∈X. The proof passes through a sequential version of being almost square which we call being 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 c0 if and only if there exists an equivalent renorming |⋅| on X for which there exists h∈SX⁎⁎⁎⁎ such that |x+h|=max⁡{|x|,1} for every x∈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∈SX⁎⁎⁎⁎ such that ‖x+h‖=max⁡{‖x‖,1} for all x∈X.