Fourier algebras on locally compact hypergroups

In the present paper we introduce a new definition for the Fourier space A(K) of a locally compact Hausdorff hypergroup K and prove that it is a Banach subspace of B(K). This definition coincides with that of Amini and Medghalchi in the case where K is a tensor hypergroup, and also with that of Vrem which is given only for compact hypergroups. We prove that A(p)(K)* = PMq(K), where q is the exponent conjugate to p, in particular A(K)* = VN(K). Also we show that for Pontryagin hypergroups, A(K) = L-2(K) * L-2(K) = F(L-1 ((K) over cap)), where F stands for the Fourier transform on (K) over cap. Furthermore there is an equivalent norm on A(K) which makes A(K) into a Banach algebra isomorphic with L-1(K) over cap. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim