Turbulence, orbit equivalence, and the classification of nuclear C^∗-algebras

TY - JOUR

T1 - Turbulence, orbit equivalence, and the classification of nuclear C∗-algebras

AU - Farah, Ilijas

AU - Toms, Andrew

AU - Törnquist, Asger Dag

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

U2 - 10.1515/crelle-2012-0053

DO - 10.1515/crelle-2012-0053

M3 - Journal article

SN - 0075-4102

SP - 101

EP - 146

JO - Journal fuer die Reine und Angewandte Mathematik

JF - Journal fuer die Reine und Angewandte Mathematik

IS - 688

ER -