Homomorphisms of \(l^1\)-algebras on signed polynomial hypergroups
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 \textit{W. R. Bloom} and \textit{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 \textit{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.
scientific article; zbMATH DE number 5702409