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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 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:

43A62 Harmonic analysis on hypergroups
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46H20 Structure, classification of topological algebras
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