A classification of anomalous actions through model action absorption

TL;DR

The paper develops a framework to classify anomalous actions with the Rokhlin property on C$^*$-algebras by bootstraping from Izumi’s Rokhlin-based group-action classification. It constructs model $(G,\omega)$-actions on the UHF algebra $M_{|G|^\infty}$ with Rokhlin property and proves a model-action absorption result: Rokhlin anomalous actions on $M_{|G|^\infty}$-stable algebras are cocycle conjugate to the action tensoring with the model action. An abstract classification lemma then reduces the problem to invariants provided by a functor $\Lambda$ and the anomaly $o(\alpha,u)$, yielding concrete classifications for Kirchberg algebras in the UCT class, unital TAF algebras, and the Razak–Jacelon algebra; analogous results are obtained for $O_2$-based settings. The work further connects to AF-actions via the AF-invariants of fusion categories and exhibits the existence of AF $\omega$-anomalous actions $\theta_G^{\omega}$ on $M_{|G|^\infty}$, clarifying when Rokhlin anomalous actions are AF-actions. Overall, the paper provides a unified, scalable approach to quantum symmetries beyond group actions, with explicit classification results tied to $K$-theory, $Z^3(G,\mathbb{T})$-anomalies, and fusion-category invariants.

Abstract

We discuss a strategy for classifying anomalous actions through model action absorption. We use this to upgrade existing classification results for Rokhlin actions of finite groups on C$^*$-algebras, with further assuming a UHF-absorption condition, to a classification of anomalous actions on these C$^*$-algebras.

Theorems & Definitions (42)

• Theorem A
• Theorem B
• Definition A
• Remark B
• Definition C
• Remark D
• Definition E
• Definition F
• Definition G
• Proposition A