A Characterization of Weakly Lindelöf Determined Banach Spaces

2000 Mathematics Subject Classification: 46B26, 46B03, 46B04.

We prove that a Banach space X is weakly Lindelöf determined if (and only if) each non-separable Banach space isomorphic to a complemented subspace of X has a projectional resolution of the identity. This answers a question posed by S. Mercourakis and S. Negrepontis and yields a converse of Amir-Lindenstrauss’ theorem. We also prove that a Banach space of the form C(K) where K is a continuous image of a Valdivia compactum is weakly Lindelöf determined if (and only if) each non-separable Banach space isometric to a subspace of C(K) has a projectional resolution of the identity.

This work was partially supported by Research grants GAUK 1/1998, GAUK 160/1999, GA CR 201/00/1466 and MSM 113200007.