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 )* = PM q ( 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 ( $ \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 L 1 ( $ \hat K $ ). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)