Fourier algebras on locally compact hypergroups

Abstract

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 Ap (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) = L2(K) * L2(K) = F (L1($ \hat K $)), where F stands for the Fourier transform on $ \hat K $. Furthermore there is an equivalent norm on A (K) which makes A (K) into a Banach algebra isomorphic with L1($ \hat K $). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)