Projective resolutions, Markushevich bases, and equivalent norms (Q797096)

Let \(\omega\) (\(\Omega)\) be the first infinite (uncountable) ordinal and let dens X be the smallest cardinal c for which the Banach space X has a dense subset of cardinality c. Given an ordinal \(\alpha\), denote by \({\bar\alpha}\) its cardinal. The main result of this paper is the following theorem: Let \(P_{\beta}\), \(Q_{\beta}\), \(\omega \leq \beta \leq \Omega\), be two projective resolutions of the Banach space X (dens X\(=\Omega)\) such that \((*)\quad X^*=\cup_{\beta<\Omega}P^*_{\beta}X^*\). Then for each \(\beta_ 0<\Omega\) there exists an ordinal \(\gamma\) such that \(P_{\gamma}=Q_{\gamma}\), \(\beta_ 0<\gamma<\Omega.\) It is clear that if property (*) is fulfilled for one projective resolution then it is fulfilled for all projective resolutions. But, the author proves (Th. 2) that the property (*) is not allways fulfilled even for Banach spaces with M-bases; for instance the space \(C_ 0(\Omega)\). From the main theorem, certain existence results of M-bases are derived. Thus, one of this results, which at the same time solves a problem from \textit{I. Singer's} book ''Bases in Banach Spaces. II.'' (1981; Zbl 0467.46020), (Problem 20.8, p. 837) says: If a Banach space has a locally uniformly convex norm and a weak* angelic dual, then projection resolutions and M-bases do not exist.