Conjugacy and cocycle conjugacy of automorphisms of \(\mathcal{O}_{2}\) are not Borel

The group of automorphisms of the Cuntz algebra \(\mathcal{O}_{2}\) is a Polish group with respect to the topology of pointwise convergence in norm. Our main result is that the relations of conjugacy and cocycle conjugacy of automorphisms of \(\mathcal{O}_{2}\) are complete analytic sets and, in par...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Gardella, Eusebio, Martino Lupini
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.04.2014
Subjects
Algebra
Automorphisms
Classification
Completeness
Complexity theory
Isomorphism
Theorems
Topology
Online AccessGet full text

Cover

More Information
Summary:The group of automorphisms of the Cuntz algebra \(\mathcal{O}_{2}\) is a Polish group with respect to the topology of pointwise convergence in norm. Our main result is that the relations of conjugacy and cocycle conjugacy of automorphisms of \(\mathcal{O}_{2}\) are complete analytic sets and, in particular, not Borel. Moreover, we show that from the point of view of Borel complexity theory, classifying automorphisms of \(\mathcal{O}_{2}\) up to conjugacy or cocycle conjugacy is strictly more difficult than classifying up to isomorphism any class of countable structures with Borel isomorphism relation. In fact the same conclusions hold even if one only considers automorphisms of \(\mathcal{O}_{2}\) of a fixed finite order. In the course of the proof we will show that the relation of isomorphism of Kichberg algebras (with trivial \(K_{1}\)-group and satisfying the Universal Coefficient Theorem) is a complete anaytic set. Moreover, it is strictly more difficult to classify Kirchberg algebras (with trivial \(K_{1}\)-group and satisfying the Universal Coefficient Theorem) than classifying up to isomorphism any class of countable structures with Borel isomorphism relation.
ISSN:2331-8422