Tsirelson-like spaces and complexity of classes of Banach spaces

Abstract

Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_0$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are ${\mathbf{\Pi }}_2^1$-complete.

References in corpus (8)

• A proof of uniqueness of the Gurarii space
• Quantitative Dunford-Pettis property
• The complexity of classifying separable Banach spaces up to isomorphism
• On a difference between quantitative weak sequential completeness and the quantitative Schur property
• New proofs of Rosenthal's ${\ell }^1$--theorem and the Josefson--Nissenzweig theorem
• On the structure of separable ${\mathcal{L}}_{\infty }$-spaces
• On the complexity of some classes of Banach spaces and non-universality
• Duality on Banach spaces and a Borel parametrized version of Zippin's theorem

Cited by in corpus (5)

• Polish spaces of Banach spaces
• Complexity of distances: Theory of generalized analytic equivalence relations
• Complexity of distances: Reductions of distances between metric and Banach spaces
• The complexity of some ordinal determined classes of operators
• Universality, complexity and asymptotically uniformly smooth Banach spaces