A missing theorem on dual spaces

TL;DR

This paper investigates whether complex Banach spaces with real preduals necessarily have complex preduals, providing affirmative answers using operator space methods for both Banach and operator spaces.

Contribution

It proves that complex Banach spaces with real preduals also have complex preduals, using novel operator space techniques to solve longstanding questions.

Findings

Complex Banach spaces with real preduals have complex preduals.

Operator space methods effectively address predual existence questions.

Results extend to spaces of operators on Hilbert 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.