On the fourier algebra of certain hypergroups

Abstract

We consider certain classes of hypergroups and address some problems regarding their Fourier algebras, including zero product preserving maps, local derivations and weak amenability. The key tool in our approach is to show that the Fourier algebra of such hypergroups has the so called property (????). This property has already been shown to be very effective in solving problems related to zero product preserving maps and local derivations provided that some other conditions are satisfied. Some examples of groups and hypergroups, the Fourier algebras of which have the property (????) satisfying all those conditions are given. Our examples are the class of almost abelian groups and double coset hypergroups of Gelfand pairs. We also prove that if H is the double coset hypergroup of a Gelfand pair, then A(H) is weakly amenable. We provide some results on the direct product of commutative (regular) Fourier hypergroups.