Centers of hypergroups

Abstract

This paper initiates a study of Z-hypergroups, that is, commutative topological hypergroups K such that K / Z K/Z is compact where Z denotes the maximum subgroup (equivalently, the center) of K. The character hypergroup K ∧ {K^\wedge } is studied and its connection with the locally compact abelian group Z ∧ {Z^\wedge } is given. Each Z-group is shown to correspond in a natural way to a Z-hypergroup. It is observed that the dual of a Z-group is itself a hypergroup. The basic orthogonality relations on Z-groups due to S. Grosser and M. Moskowitz are shown to hold for most Z-hypergroups. Some results on measure algebras of compact hypergroups due to C. F. Dunkl are extended to a class of noncompact hypergroups.