AT-nature and equivalence of tensor-extended actions versus AT-construction

Determine whether, for a unitary fusion category C with a simple stationary AF-action arising from module data (M, m, X) such that the fusion matrix of X on M is invertible over the integers, the action on the C*-algebra B ≅ D ⊗ A obtained by tensor extension with a classifiable C*-algebra D satisfying K0(D) ≅ K1(D) ≅ Z and having a unique trace is an AT-action; and ascertain whether this tensor-extended action is equivalent, as a C-action, to the AT-action on B produced by the inductive-limit circle-algebra construction in Theorem 2.*.

Background

The paper constructs AT-actions of fusion categories on AT-algebras via an inductive limit of circle algebras (Theorem 2.). It also observes a second way to obtain an action: start with the AF-action on an AF-algebra A coming from simple stationary data (M, m, X) with an invertible fusion matrix, and tensor with a classifiable C-algebra D having K0(D) ≅ K1(D) ≅ Z and unique trace. Classification then gives an isomorphism B ≅ D ⊗ A.

However, because D itself is not AT, it is unclear whether the tensor-extended action on B is an AT-action in the sense of arising from an inductive system of circle-algebra building blocks. Moreover, even if both actions live on isomorphic AT-algebras B, it is unknown whether the two actions are equivalent as actions of the fusion category. The authors highlight this as a natural test case for future classification results of fusion category actions beyond the AF-setting.