A Plancherel Theorem On a Noncommutative Hypergroup

Abstract

Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G. (G, K) is a Gelfand pair if Mc(G//K), the algebra of measures with compact support on the double coset G//K, is commutative for the convolution. In this paper, assuming that (G, K) is a Gelfand pair, we define and study a Fourier transform on G and then establish a Plancherel theorem for the pair (G, K).