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.

arXiv comment52 pages, comments welcome!

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• Quantum graphs, subfactors and tensor categories I
• The stable uniqueness theorem for unitary tensor category equivariant KK-theory
• Separable algebras in multitensor C${}^{\ast }$-categories are unitarizable
• A classification of anomalous actions through model action absorption
• Discrete Inclusions of C*-algebras