Turbulence, orbit equivalence, and the classification of nuclear C∗-algebras
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Turbulence, orbit equivalence, and the classification of nuclear C∗-algebras. / Farah, Ilijas; Toms, Andrew; Törnquist, Asger Dag.
I: Journal fuer die Reine und Angewandte Mathematik, Nr. 688, 2014, s. 101-146.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 O2 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 O2-stability and embedding theorems. We also find a C*-algebraic witness for a 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 O2 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 O2-stability and embedding theorems. We also find a C*-algebraic witness for a Kσ hard equivalence relation.
U2 - 10.1515/crelle-2012-0053
DO - 10.1515/crelle-2012-0053
M3 - Journal article
SP - 101
EP - 146
JO - Journal fuer die Reine und Angewandte Mathematik
JF - Journal fuer die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 688
ER -
ID: 135505703