Regular inclusions of simple unital $C^{\ast }$-algebras

Abstract

We prove that an inclusion $\mathcal{B}\subset \mathcal{A}$ of simple unital $C^{\ast }$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(\alpha ,\sigma )$ on the intermediate $C^{\ast }$-subalgebra $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer ${\mathcal{C}}_{\mathcal{A}}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $\alpha$-invariant and $(\mathcal{B}\subset \mathcal{A})\cong (\mathcal{B}\subset \mathcal{C}{⋊}_{\alpha ,\sigma }^{r}G)$. 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 $2$.

16 pages