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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 = 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:
43A62Hypergroups (abstract harmonic analysis)
43A22Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A20L 1 -algebras on groups, semigroups, etc.
46H20Structure and classification of topological algebras