Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries

Abstract

Let $B \subset A$ be a depth $2$ inclusion of simple unital $C^*$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B' \cap A_1$ carries a canonical structure of a weak $C^*$-Hopf algebra. Furthermore, we construct an action of this weak $C^*$-Hopf algebra on $A$ for which $B$ is precisely the fixed-point subalgebra, and we prove that the first basic construction $A_1$ is isomorphic to the crossed product $A \rtimes (B' \cap A_1)$. This provides a $C^*$-algebraic counterpart of the duality between depth $2$ subfactors and weak Hopf algebra symmetry, extending the Ocneanu-Nikshych-Vainerman theory beyond the $II_1$ factor setting.