One-dimensional hypergroups (Q1123342)

A hypergroup is a locally compact space K equipped with a multiplication * on the space \({\mathcal M}^ b(K)\) of bounded measures on K. The author investigates hypergroups on one dimensional spaces, \(K={\mathbb{R}},{\mathbb{T}},{\mathbb{R}}_+\), or [0,1]. Several nice answers are given to very natural questions. It is proved that if \(K={\mathbb{R}}\) or \({\mathbb{T}}\), then (K,*) is isomorphic to the hypergroup induced by the usual group structure on \({\mathbb{R}}\) or \({\mathbb{T}}\). If \(K={\mathbb{R}}_+\) or [0,1], then the hypergroup (K,*) is commutative. The hypergroups ([0,1],*) such that the support of \(\epsilon_ x*\epsilon_ y\) contains at most two points for every x, y, is isomorphic with the double coset space \({\mathbb{T}}\oplus {\mathbb{Z}}_ 2//{\mathbb{Z}}_ 2\). For a Sturm-Liouville hypergroup on \(({\mathbb{R}}_+,*)\) the support of \(\epsilon_ x*\epsilon_ y\) is the interval \([| x-y|,x+y]\).