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The complexity of classifying separable Banach spaces up to isomorphism. (English) Zbl 1234.46008

Classifying Banach spaces up to linear isomorphism is notoriously difficult, and it is important to give a precise mathematical content to such a statement. A first step in this direction was done by B. Bossard who showed in his PhD Thesis (published in 2002) that the collection of separable Banach spaces can be equipped with a natural structure of standard Borel space, and that when this is done, the isomorphism equivalence relation is analytic non Borel. The present work goes further, and provides a final satisfactory answer to the initial query.
It is by now classical that equivalence relations on standard Borel spaces can be compared through Borel ordering. Roughly speaking, a relation \(R_1\) will be less complex than a relation \(R_2\) if \(R_1\)-equivalence reduces to \(R_2\)-equivalence in a constructive (that is, Borel) manner. For instance, Stone duality allows to classify countable Boolean algebras up to isomorphisms by compact metric spaces up to homeomorphisms. The main result of the present work is that every analytic equivalence relation is Borel-reducible to the isomorphism relation between separable Banach spaces. In other words, the complexity of the linear isomorphism relation is maximal among all analytic equivalence relations. Besides its intrinsic interest, this result also provides the first “concrete” example of a maximal analytic equivalence relation. The proofs use refined tools from descriptive set theory and geometry of Banach spaces, and in particular the deep amalgamation technique invented by S. Argyros and P. Dodos in 2007.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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