On the existence and uniqueness of complex structure and spaces with “few” operators

Abstract

We construct a 2 n 2n -dimensional real normed space whose (Banach-Mazur) distance to the set of spaces admitting complex structure is of order n 1 / 2 {n^{1/2}} , and two complex n n -dimensional normed spaces which are isometric as real spaces, but whose complex Banach-Mazur distance is of order n n . Both orders of magnitude are the largest possible. We also construct finite-dimensional spaces with the property that all “well-bounded” operators on them are “rather small” (in the sense of some ideal norm) perturbations of multiples of identity. We also state some “metatheorem”, which can be used to produce spaces with various pathological properties.