Authors: Farah, Ilijas 
Toms, Andrew
Törnquist, Asger
Title: Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
Journal: Journal fur die Reine und Angewandte Mathematik
Issue: 688
First page: 101
Last page: 146
Issue Date: 17-Mar-2014
Rank: M21a
ISSN: 0075-4102
DOI: 10.1515/crelle-2012-0053
Abstract: 
We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C*-algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra ω2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C*-algebras by K-theory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C*-algebras, we prove that many standard C*-algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg's ω2$ -stability and embedding theorems. We also find a C*-algebraic witness for a K σ Kσ hard equivalence relation.
Publisher: De Gruyter
Project: NSF, Grant DMS-0969246
The Austrian Science Fund FWF, Grant no. P 19375-N18
Danish Council for Independent Research (Natural Sciences), Grant no. 10-082689/FNU
Marie Curie re-integration, Grant no. IRG-249167

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