Point Derivations on the l-Algebra of Polynomial Hypergroups

Polynomial hypergroups are a very interesting class of hypergroups with a great variety of examples which are quite different from groups. So the L-algebras of hypergroups have properties very distinguished to the L-algebras of groups, in particular in the context of amenability and related conditions. Being amenable the L-algebra of an abelian group does not possess any non-zero bounded point derivation, see e.g. [5, p.214]. We will show that for the L-algebra of hypergroups, indeed we restrict ourselves to polynomial hypergroups, the situation is rather different. To have a good reference and for the sake of completeness we recall shortly the basic facts for polynomial hypergroups. For more details and the proofs we refer to [14] and [15].