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Homomorphisms of $l^1$-algebras on signed polynomial hypergroups. (English) Zbl 1191.43005
Let $\{R_n\}$ and $\{P_n\}$ be two polynomial systems which induce signed polynomial hypergroup structures on $N_0$. The paper under review investigates when the Banach algebra $l^1(N_0, h^R)$ can be continuously embedded into or is isomorphic to $l^1(N_0, h^P)$. 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 {\it W. R. Bloom} and {\it 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 {\it 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.

43A62Hypergroups (abstract harmonic analysis)
43A22Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A20$L^1$-algebras on groups, semigroups, etc.
46H20Structure and classification of topological algebras
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