Depth 2 inclusions of simple C∗-algebras and their weak C∗-Hopf algebra symmetries

Let [Formula: see text] be a depth [Formula: see text] inclusion of simple unital [Formula: see text]-algebras with a conditional expectation of index-finite type. We show that the second relative commutant [Formula: see text] carries a canonical structure of a weak [Formula: see text]-Hopf algebra. Furthermore, we construct an action of this weak [Formula: see text]-Hopf algebra on [Formula: see text] for which [Formula: see text] is precisely the fixed-point subalgebra, and we prove that the first basic construction [Formula: see text] is isomorphic to the crossed product [Formula: see text]. This provides a [Formula: see text]-algebraic counterpart of the duality between depth 2 subfactors and weak Hopf algebra symmetry, extending the Ocneanu–Nikshych–Vainerman theory beyond the [Formula: see text] factor setting.