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Point derivations on the $L^1$-algebra of polynomial hypergroups. (English) Zbl 1167.43007
Summary: We investigate whether the $L^1$-algebra of polynomial hypergroups has non-zero bounded point derivations. We show that the existence of such point derivations heavily depends on growth properties of the Haar weights. Many examples are studied in detail. We can thus demonstrate that the $L^1$-algebras of hypergroups have properties (connected with amenability) that are very different from those of groups.

##### MSC:
 43A62 Hypergroups (abstract harmonic analysis) 43A07 Means on groups, semigroups, etc.; amenable groups 43A15 $L^p$-spaces and other function spaces on groups, semigroups, etc. 46H20 Structure and classification of topological algebras
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