Weak Quantum Hypergroups from Finite Index C*-Inclusions

Abstract We study a finite index inclusion $B \subset A$ of simple unital $C^{*}$-algebras and construct a canonical completely positive coproduct on the second relative commutant $B^{\prime} \cap A_{1}$, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman [12]. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth $2$ setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically $C^{*}$-algebraic framework for generalized quantum symmetries associated with finite index inclusions.