A superreflexive Banach space which does not admit complex structure

Abstract

We construct an infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space. We also construct a variant of Bourgain’s example of a complex Banach space with nonunique complex structure and state a number of open problems about structure of Banach spaces and their linear groups.