A missing theorem on dual spaces

Abstract

We answer in the affirmative the surprisingly difficult questions: If a complex Banach space possesses a real predual X, then is X a complex Banach space? If a complex Banach space possesses a real predual, then does it have a complex predual? We also answer the analogous questions for operator spaces, that is spaces of operators on a Hilbert space, up to complete isometry. Indeed we use operator space methods to solve the Banach space question above.

Error in early lemma (Proposition 2.3) endangers Theorem 4.1. Result is valid assuming the conclusions of Proposition 2.3 for the map q in Theorem 4.1