We prove that an inclusion B ⊂ A of simple unital C*-algebras with a finiteindex conditional expectation is regular if and only if there exists a finite group G that admits a cocycle action (α, σ) on the intermediate C*-subalgebra C generated by B and its centralizer CA(B) such that B is outerly α-invariant and (B ⊂ A) ∼= (B ⊂ C ⋊r (α,σ) G). Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most 2.
Bakshi et al. (Wed,) studied this question.