Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

Abstract

There exists a real hereditarily indecomposable Banach space X = X ( C ) (respectively X = X ( H ) ) such that the algebra L ( X ) / S ( X ) is isomorphic to C (respectively to the quaternionic division algebra H ). Up to isomorphism, X ( C ) 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 [J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (2) (1986) 221–226; S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1) (1986) 339–353; S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437–444], and proves that a theorem of G. Godefroy and N.J. Kalton [G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121–141] about isometric embeddings of separable real Banach spaces does not extend to the complex case. The quaternionic example X ( H ) , on the other hand, has unique complex structure up to isomorphism; other examples with a unique complex structure are produced, including a space with an unconditional basis and non-isomorphic to l 2 . This answers a question of S. Szarek in [S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437–444].