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Amenability and weak amenability of ${l}^{1}$-algebras of polynomial hypergroups. (English) Zbl 1126.43003
It is well known that all commutative hypergroups are amenable in the sense that an invariant mean exists. However, in contrast to the group case, there exist commutative hypergroups for which other versions of amenability fail. For instance, it is well-understood when Reiter’s condition (P2) fails. In the paper under review, the amenability of the Banach-$*$-algebra ${l}^{1}$ is studied for polynomial hypergroups. It turns out that most known polynomial hypergroups are not amenable in this sense, and that the only known positive example appears for the hypergroup associated with Chebychev polynomials of the first kind.

##### MSC:
 43A62 Hypergroups (abstract harmonic analysis) 46H20 Structure and classification of topological algebras 43A07 Means on groups, semigroups, etc.; amenable groups 46H25 Normed modules and Banach modules, topological modules
##### Keywords:
polynomial hypergroups; amenability