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 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. 
Eusebio Gardella and Martino Lupini
math.OA