K-theoretic classification of inductive limit actions of fusion categories 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.