Let and be two polynomial systems which induce signed polynomial hypergroup structures on . The paper under review investigates when the Banach algebra can be continuously embedded into or is isomorphic to . Certain sufficient conditions on the connection coefficients given by , for the existence of such an embedding or isomorphism are given. These results are also applied to obtain amenability properties of the -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 -algebras of polynomial hypergroups.