Almost square dual Banach spaces

Abstract

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on ${\ell }_{\infty }.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.

Added a new result (Theorem 3.7) that there exists an equivalent renorming of c_0 which is M-embedded and whose bidual is ASQ. Added a couple of new paragraphs and fixed some typos