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Homomorphisms of ${l}^{1}$-algebras on signed polynomial hypergroups. (English) Zbl 1191.43005

Let $\left\{{R}_{n}\right\}$ and $\left\{{P}_{n}\right\}$ be two polynomial systems which induce signed polynomial hypergroup structures on ${N}_{0}$. The paper under review investigates when the Banach algebra ${l}^{1}\left({N}_{0},{h}^{R}\right)$ can be continuously embedded into or is isomorphic to ${l}^{1}\left({N}_{0},{h}^{P}\right)$. Certain sufficient conditions on the connection coefficients ${c}_{n,k}$ given by ${R}_{n}={\sum }_{k=0}^{n}{c}_{nk}{P}_{k}$, for the existence of such an embedding or isomorphism are given. These results are also applied to obtain amenability properties of the ${l}^{1}$-algebras induced by Bernstein-Szegő and Jacobi polynomials.

The previous related investigations can be found in W. R. Bloom and M. E. Walter’s work [J. Aust. Math. Soc., Ser. A 52, No. 3, 383–400 (1992; Zbl 0776.43001)], which was only concerned with the isometric isomorphisms of hypergroups. For more recent works, see R. Lasser’s articles [Stud. Math. 182, No. 2, 183–196 (2007; Zbl 1126.43003); Colloq. Math. 116, No. 1, 15–30 (2009; Zbl 1167.43007)], which studied the amenability of ${l}^{1}$-algebras of polynomial hypergroups.

##### MSC:
 43A62 Hypergroups (abstract harmonic analysis) 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A20 ${L}^{1}$-algebras on groups, semigroups, etc. 46H20 Structure and classification of topological algebras
##### Keywords:
Banach algebra homomorphism; hypergroup; amenability