On the classification of positions and of complex structures in Banach spaces

2017/01/16 by Anisca, Razvan, Ferenczi, Valentin, Moreno, Yolanda
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1701.04263

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 E₀ 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 E₁ 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_{\rm max}; d) there exists a subspace of L_p, 1 ≤ p <2, on which isomorphism between complex structures reduces E₁ and therefore is not reducible to an orbit relation induced by the action of a Polish group.

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