On a categorical framework for classifying C⁎-dynamics up to cocycle conjugacy

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

“…In this section we shall consider a slight variant of Szabó's approximate cocycle intertwining argument in [59] (see also [9]). Throughout this section, we assume that A and B are non-unital C * -algebras.…”

“…Remark 5.1. In [59], Szabó defined cocycle morphisms and proper cocycle morphisms for (twisted) group actions on C * -algebras. Furthermore, he provided a categorical framework for the classification of group actions on C * -algebras up to cocycle conjugacy.…”

“…Our main technical tool of the proof of the theorem above is Szabó's approximate cocycle intertwining argument in [59] (see also [9]). This is an equivariant version of Elliott's approximate intertwining argument [7].…”

The stable uniqueness theorem for equivariant Kasparov theory

“…Since the methodology of Izumi-Matui is closely tied to the Rokhlin property and the Evans-Kishimoto intertwining argument [12], it is increasingly difficult to implement for more complicated groups, and in fact is too restrictive for groups with torsion. Recently, the second named author proposed a categorical framework for C * -dynamics [52] to classify group actions up to cocycle conjugacy based on an Elliott intertwining argument that is intended to provide an alternative to the one of Evans-Kishimoto. In this framework, arrows between actions of C * -algebras are so-called cocycle morphisms. In analogy to ordinary C * -algebra classification, it is an important intermediate step to solve the uniqueness problem, i.e., to determine in terms of classifying invariants when two cocycle morphisms are approximately/asymptotically unitarily equivalent; for more details see [52].…”

“…Recently, the second named author proposed a categorical framework for C * -dynamics [52] to classify group actions up to cocycle conjugacy based on an Elliott intertwining argument that is intended to provide an alternative to the one of Evans-Kishimoto. In this framework, arrows between actions of C * -algebras are so-called cocycle morphisms. In analogy to ordinary C * -algebra classification, it is an important intermediate step to solve the uniqueness problem, i.e., to determine in terms of classifying invariants when two cocycle morphisms are approximately/asymptotically unitarily equivalent; for more details see [52]. Since it was also argued that equivariant KK-theory can be viewed as a (bi-)functor on this enlarged category, it is natural expect that it ought to represent one of the key obstructions to solving said existence/uniqueness problem.…”