Positive characters on commutative hypergroups and some applications (Q1123341)

Let (K,*) be a commutative hypergroup with Haar measure m and associated Plancherel measure \(\pi\) on the dual \(\hat K.\) The author proves the existence of a unique positive character \(\alpha_ 0\in \sup p \pi\). With the aid of the character \(\alpha_ 0\) (or any other positive character on K), the author defines a new hypergroup structure (K,\(\circ)\) on K, which is connected with (K,*) in a canonical way. Using this technique of modifying the convolution structure of a hypergroup, new examples of hypergroups are constructed and relations between well known examples are obtained; e.g. the double coset hypergroups SL(2,\({\mathbb{C}})/SU(2)\) and I(3)/SO(3) are connected in this way (I(3) denotes the group of isometries on \({\mathbb{R}}^ 3)\). If (K,\(\circ)\) is the modified hypergroup obtained by using the character \(\alpha_ 0\) then the hypergroup (K,\(\circ)\) satisfies \(1\in \sup p \pi_{\alpha_ 0}\), where \(\pi_{\alpha_ n}\) is the Plancherel measure on the dual of (K,\(\circ).\) The existence of a positive character \(\alpha_ 0\in \sup p \pi\) is used to give a short proof of the transience of random walks and convolution semigroups on hypergroups satisfying \(1\in \sup p \pi\). The above results are illustrated for a class of hypergroups on the half real line \(R_+\).