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Journal of the London Mathematical Society Advance Access originally published online on January 6, 2009
Journal of the London Mathematical Society 2009 79(2):323-345; doi:10.1112/jlms/jdn068
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© 2009 London Mathematical Society

The complexity of classifying separable Banach spaces up to isomorphism

Valentin Ferenczi

Équipe d’Analyse Fonctionnelle
Université Pierre et Marie Curie – Paris 6
Boîte 186
4, Place Jussieu
75252 Paris cedex 05
France
Current address:
Departamento de Matemática
Instituto de Matemática e Estatística
Universidade de São Paulo
05311-970 São Paulo, SP
Brazil
ferenczi@ime.usp.br

Alain Louveau

Équipe d’Analyse Fonctionnelle
Université Pierre et Marie Curie – Paris 6
Boîte 186
4, Place Jussieu
75252 Paris cedex 05
France
louveau@ccr.jussieu.fr

Christian Rosendal

Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
322 Science and Engineering Offices (M/C 249)
851 S. Morgan Street
Chicago, IL 60607-7045
USA

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.

Dedicated to Alekos Kechris, on the occasion of his 60th birthday

2000 Mathematics Subject Classification 46B03 (primary), 03E15 (secondary).

The third author is partially supported by NSF grant DMS 0556368.

Received September 4, 2007; revised August 29, 2008; published online January 6, 2009.


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