This is an announcement for the paper "The complexity of classifying separable Banach spaces up to isomorphism" by Valentin Ferenczi, Alain Louveau, and Christian Rosendal.

Abstract: 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.

Archive classification: Functional Analysis; Logic

Mathematics Subject Classification: 46B03; 03E15

The source file(s), ComplexityIsomorphism14.tex: 82408 bytes, is(are) stored in gzipped form as 0610289.gz with size 25kb. The corresponding postcript file has gzipped size 101kb.

Submitted from: rosendal@math.uiuc.edu

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