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

• Published in: Journal of functional analysis Vol. 280; no. 8; p. 108927
• Main Author: Szabó, Gábor
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2021
• Subjects: Elliott intertwining; Equivariant Kasparov theory; Noncommutative [formula omitted]-dynamical systems; Twisted actions; Noncommutative C⁎-dynamical systems; Elliott intertwining; 46L55; Twisted actions; Equivariant Kasparov theory
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2021.108927

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.