Operator Algebraic Quantum Hypergroups

Abstract

We develop a theory of operator algebraic quantum hypergroups and their representations, formulated using a special case of the algebraic quantum hypergroup framework of van Daele and Delvaux. As part of the axioms, every operator algebraic quantum hypergroup admits a dual object of the same type. We prove that a certain universal completion in the compact case satisfies the axioms of an earlier notion of a compact quantum hypergroup in the sense of the second author and Chapovsky. As part of this theory, we discuss their (co)amenability properties. We consider examples induced by conditional expectations of algebraic quantum groups satisfying certain properties, which include quantum hypergroups corresponding to double cosets and automorphisms of quantum groups, and quantum hypergroups induced by representation categories of Kac type compact quantum groups.