K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

TL;DR

This paper develops a K-theoretic invariant for classifying actions of fusion categories on AF-algebras, providing a complete classification for inductive limits of finite-dimensional actions, including finite group actions.

Contribution

It introduces a new K-theoretic invariant that fully classifies inductive limit actions of fusion categories on AF-algebras, extending to finite group actions.

Findings

Complete invariant for inductive limit actions of fusion categories

Classification of finite depth, strongly AF-inclusions

Applicable to finite group actions on AF-algebras

Abstract

We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras.