Actions of tensor categories on Kirchberg algebras

TL;DR

This paper characterizes when Pimsner algebras are simple and uses this to systematically construct outer actions of unitary tensor categories on Kirchberg algebras, including all countable categories and specific twisted Cuntz algebras.

Contribution

It provides a criterion for simplicity of Pimsner algebras and demonstrates how to produce outer tensor category actions on Kirchberg algebras, including new examples involving 3-cocycle twists.

Findings

Every countable unitary tensor category admits an outer action on the Cuntz algebra 2.

The realizability of modules over fusion rings as K-groups is generically true for unitary fusion categories.

Constructed actions on Cuntz algebras of 3-cocycle twists for all cohomological classes, answering Izumi's question.

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.