A converse to Amir-Lindenstrauss theorem in complex Banach spaces

Abstract

We show that a complex Banach space is weakly Lindeloef determined if and only if the dual unit ball of any equivalent norm is weak* Valdivia compactum. We deduce that a complex Banach space X is weakly Lindeloef determined if and only if any nonseparable Banach space isomorphic to a complemented subspace of X admits a projectional resolution of the identity.

These results complete the previous ones on real spaces.

Keywords

• converse
• Amir-Lindenstrauss
• theorem
• complex
• Banach
• spaces