Hypergroup structures arising from certain dual objects of a hypergroup

Abstract

In the present paper hypergroup structures are investigated on distinguished dual objects related to a given hypergroup $K$, especially to a semi-direct product hypergroup $K = H \rtimes_\alpha G$ defined by an action $\alpha$ of a locally compact group $G$ on a commutative hypergroup $H$. Typical dual objects are the sets of equivalence classes of irreducible representations of $K$, of infinite-dimensional irreducible representations of type I hypergroups $K$, and of quasi-equivalence classes of type $\text{II}_1$ factor representations of non-type I hypergroups $K$. The method of proof relies on the notion of a character of a representation of $K = H \rtimes_\alpha G$.