The complexity of classifying separable Banach spaces up to isomorphism

Abstract

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.

Cited by in corpus (6)

• Tsirelson-like spaces and complexity of classes of Banach spaces
• Measure reducibility of countable Borel equivalence relations
• Complexity of distances: Theory of generalized analytic equivalence relations
• Complexity of distances: Reductions of distances between metric and Banach spaces
• On the complexity of some inevitable classes of separable Banach spaces
• Borel fields and measured fields of Polish spaces, Banach spaces, von Neumann algebras and C*-algebras