This article is a continuation of a paper of the first author \cite{ F} 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 \cite{ F} and $C(K)$ examples due to Plebanek \cite{ P}. We extend results of \cite{ F} 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 \cite{ F} 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 \cite{ AM} provide examples of essentially incomparable complex structures which are not totally incomparable.
Categories: math.FA