Homomorphisms of $l^1$-algebras on signed polynomial hypergroups

Abstract

Let $\{R_n\}$ and $\{P_n\}$ be two polynomial systems which induce signed polynomial hypergroup structures on $\mathbb{N}_0.$ We investigate when the Banach algebra $l^1(\mathbb{N}_0,h^R)$ can be continuously embedded into or is isomorphic to $l^1(\mathbb{N}_0,h^P).$ We find sufficient conditions on the connection coefficients $ c_{nk} $ given by $ R_n = \sum_{k=0}^n c_{nk} P_k, $ for the existence of such an embedding or isomorphism. Finally we apply these results to obtain amenability-properties of the $l^1$-algebras induced by Bernstein-Szego and Jacobi polynomials.