Induced representations of hypergroups (Q1204313)

The abstract inducing process of M. Rieffel is used in this paper to introduce induced representations of hypergroups. Contrary to the group case, there exist representations of subhypergroups, even for commutative hypergroups, which do not admit induced representations. On the other hand, the main results of this paper state that the left regular representation \(\lambda_ H\) of a subhypergroup \(H\) of a hypergroup \(K\) is always inducible to \(K\) with \(\text{ind }\lambda_ H=\lambda_ K\), and that the property of being inducible is preserved under weak containment. As an application of these facts it is shown that each character on a subhypergroup \(H\) of a commutative hypergroup \(K\) which is contained in the support of the Plancherel measure of \(H\) can always be extended to a character on \(K\). The paper also contains hypergroup versions of the Mackey tensor product theorem as well as the Frobenius reciprocity theorem for compact hypergroups besides some further basic properties of induced representations on hypergroups.