On renormings of nonreflexive Banach spaces with preduals (Q1813748)

It is well known that if a Banach space \(X\) is isometric to a dual Banach space then there exists a projection \(P\) with norm 1 from its second dual \(X^{**}\) onto \(X\). Davis and Johnson showed that every nonreflexive Banach space can be equivalently renormed in such a way that the renormed space is not isometric to a dual space. Dulst and Singer then proved that this conclusion can be improved in the following form: Every nonreflexive Banach space \(X\) admits an equivalent norm \(\|\cdot\|\) such that for each projection \(P: X^{**}\to X\), \(\| P\|>1\) with respect to the norm. After this, Godun gave a more general result that each nonreflexive Banach space \(X\) admits an equivalent norm \(\|\cdot\|\) such that for each projection \(P: X^{**}\to X\), \(\| P\|\geq 2\) with respect to the norm. These results are all related to the existence of an equivalent norm which admits no preduals. On the contrary, we are concerned with equivalent norms which admit preduals. In this paper we consider the class of such equivalent norms and demonstrate that given a nonreflexive Banach space \(X\) ``most'' of the equivalent norms on \(X\) do not admit preduals in the above sense.