On the classification of positions and complex structures in Banach spaces

Abstract

A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The following results are obtained: a) if X is not uniformly finitely extensible, then there exists a space Y for which the relation of position of Y inside X reduces the relation E 0 and therefore is not smooth; b) the relation of position of ℓ p inside ℓ p , or inside L p , p = 2, reduces the relation E 1 and therefore is not reducible to an orbit relation induced by the action of a Polish group; c) the relation of position of a space inside another can attain the maximum complexity E max ; d) there exists a subspace of L p , 1 ≤ p < 2, on which isomorphism between complex structures reduces E 1 and therefore is not reducible to an orbit relation induced by the action of a Polish group.

There is no largest proper operator ideal

“…An example with exactly ℵ 0 complex structures is due to Cuellar [8], and one with 2 ℵ 0 and additional properties is due to Anisca [2] (it is not hard to check that the original example of Bourgain also admits 2 ℵ 0 such structures). See also [3] for considerations on the number of complex structures in the setting of complexity of equivalence relations on Polish spaces.…”