Regular inclusions of simple unital $C^*$-algebras

We prove that an inclusion $\mathcal{B} \subset \mathcal{A}$ of simple unital $C^*$-algebras with a finite-index 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 $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer $\mathcal{C}_\mathcal{A}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $α$-invariant and $(\mathcal{B} \subset \mathcal{A}) \cong ( \mathcal{B} \subset \mathcal{C}\rtimes^r_{α, σ} G)$. 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 $2$.

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