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The complexity of classifying separable Banach spaces up to isomorphism
Author(s) -
Ferenczi Valentin,
Louveau Alain,
Rosendal Christian
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn068
Subject(s) - mathematics , separable space , equivalence relation , isomorphism (crystallography) , banach space , borel equivalence relation , homeomorphism (graph theory) , pure mathematics , equivalence (formal languages) , banach manifold , lipschitz continuity , isomorphism extension theorem , metric space , relation (database) , discrete mathematics , lp space , mathematical analysis , fixed point theorem , probability measure , computer science , borel measure , chemistry , crystal structure , crystallography , picard–lindelöf theorem , database
It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, that is, that any analytic equivalence relation Borel reduces to it. This solves a problem of G. Godefroy. 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.