Real hereditarily indecomposable Banach spaces and uniqueness of complex structure

Abstract

There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the example with complex quotient space has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek from 1986. The quaternionic example, on the other hand, has unique complex structure up to isomorphism; there also exists a space with an unconditional basis, non isomorphic to $l_2$, which admits a unique complex structure. These two examples answer a question of Szarek.