The complexity of classifying separable Banach spaces up to isomorphism

Valentin Ferenczi, Alain Louveau, Christian Rosendal
2009 Journal of the London Mathematical Society  
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
more » ... 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.
doi:10.1112/jlms/jdn068 fatcat:qkqib5jxdvftlkbmg2x3vfllby