Depth 2 inclusions of simple $C^{\ast }$-algebras and their weak $C^{\ast }$-Hopf algebra symmetries

Let $B\subset A$ be a depth $2$ inclusion of simple unital $C^{\ast }$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B^{\mathrm{\prime }}\cap A_1$ carries a canonical structure of a weak $C^{\ast }$-Hopf algebra. Furthermore, we construct an action of this weak $C^{\ast }$-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 $A⋊(B^{\mathrm{\prime }}\cap A_1)$. This provides a $C^{\ast }$-algebraic counterpart of the duality between depth $2$ subfactors and weak Hopf algebra symmetry, extending the Ocneanu-Nikshych-Vainerman theory beyond the $I{I}_1$ factor setting.