Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C(K) examples due to Plebanek G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.