Actions of tensor categories on Kirchberg algebras

Abstract

We characterize the simplicity of Pimsner algebras for non-proper C*-correspondences. With the aid of this criterion, we give a systematic strategy to produce outer actions of unitary tensor categories on Kirchberg algebras. In particular, every countable unitary tensor category admits an outer action on the Cuntz algebra $\mathcal{O}_2$. We also study the realizability of modules over fusion rings as K-groups of Kirchberg algebras acted on by unitary tensor categories, which turns out to be generically true for every unitary fusion category. Several new examples are provided, among which actions on Cuntz algebras of 3-cocycle twists of cyclic groups are constructed for all possible 3-cohomological classes, thereby answering a question asked by Izumi.