Turbulence, orbit equivalence, and the classification of nuclear   C*-algebras
  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 O_2 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 O_2-stability and embedding theorems. We also find a C*-algebraic witness for a K_\sigma hard equivalence relation. 
Ilijas Farah, Asger Tornquist, Andrew S. Toms
math.OA math.LO