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

Abstract

We provide the rigorous foundations for a categorical approach to the classification of C⁎-dynamics up to cocycle conjugacy. Given a locally compact group G, we consider a category of (twisted) G-C⁎-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of G-C⁎-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C⁎-algebras. This reduces a given classification problem for C⁎-dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the Evans–Kishimoto intertwining argument in future research.