Valentin Ferenczi, Alain Louveau, Christian Rosendal
Published 2006-10-09Version 1
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.
Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces
Complexity of distances between metric and Banach spaces
The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $\log$-algebras
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