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 AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on unital AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras. Contents 18 4.1. Inductive limit C * -algebras 19 4.2. AF-actions 22 4.3. MPO symmetries of spin chains 24 4.4. Classification 27 4.5. Classification of strongly AF-inclusions 30 5. Examples 32 5.1. Torsion-free fusion categories 33 5.2. Hilb(Z/pZ)-actions 34 5.3. Computing the invariant in practice 35 Appendix A. [QSys(Hilb(Z/4Z))] 47 References 48

Aperiodic dynamical inclusions of C*-algebras

“…In this section we shall succinctly introduce the necessary definitions and relevant tools, while we also establish the notation. A more involved description of these concepts can be found in [CHPJP22,CHPJ22], and for general references on 2-categories see [JY21]. We will assume that all hom 1-categories in our C* 2-categories are Cauchy complete.…”