Banach Journal of Mathematical Analysis

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

Rupert Lasser and Eva Perreiter

Full-text: Open access


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.

Article information

Banach J. Math. Anal., Volume 4, Number 2 (2010), 1-10.

First available in Project Euclid: 7 February 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A62: Hypergroups
Secondary: 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

Banach algebra homomorphism hypergroup amenability


Lasser , Rupert; Perreiter , Eva. Homomorphisms of $l^1$-algebras on signed polynomial hypergroups. Banach J. Math. Anal. 4 (2010), no. 2, 1--10. doi:10.15352/bjma/1297117237.

Export citation


  • R. Askey, Orthogonal polynomials and special functions, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, 1975.
  • R. Askey and G. Gasper, Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients, Proc. Camb. Philos. Soc. 70 (1971), 243–255.
  • W.R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter, Berlin, 1995.
  • W.R. Bloom and M.E. Walter, Isomorphisms of hypergroups, J. Aust. Math. Soc., Ser. A 52 (1992), 383–400.
  • H.G. Dales, Banach algebras and automatic continuity, London Mathematical Society, Oxford, 2000.
  • F. Filbir, R. Lasser and R. Szwarc, Reiter's condition $P_1$ and approximate identities for polynomial hypergroups, Monatsh. Math. 143 (2004), 189–203.
  • V. Hösel and R. Lasser, Approximation with Bernstein-Szegő polynomials, Numer. Funct. Anal. Optim. 27 (2006), 377–389.
  • E. Kaniuth, A.-T. Lau and J. Pym, On $\varphi $-amenability of Banach algebras, Math. Proc. Camb. Philos. Soc. 144 (2008), 85–96.
  • R. Lasser, Fourier-Stieltjes transforms on hypergroups, Analysis (Munich) 2 (1982), 281–303.
  • ––––, Orthogonal polynomials and hypergroups, Rend. Mat. Appl., VII. Ser. 3 (1983), 185–209.
  • ––––, Orthogonal polynomials and hypergroups II: The symmetric case, Trans. Amer. Math. Soc. 341 (1994), 749–770.
  • ––––, Amenability and weak amenability of $l^1$-algebras of polynomial hypergroups, Studia Math. 182 (2007), 183–196.
  • ––––, Point derivations on the $L^1$-algebra of polynomial hypergroups, Colloq. Math. 116 (2009), 15–30.
  • R. Lasser and M. Rösler, A note on property (T) of orthogonal polynomials, Arch. Math. (Basel) 60 (1993), 459–463.
  • A.W. Parr, Signed Hypergroups, Ph.D. thesis, Graduate School of the University of Oregon, Department of Mathematics, 1997.
  • M. Rösler, Convolution algebras which are not necessarily positivity-preserving, Applications of hypergroups and related measure algebras, 299–318, Connett, William C. (ed.) et al., American Mathematical Society, Providence, 1995.
  • ––––, On the dual of a commutative signed hypergroup, Manuscripta Math. 88 (1995), 147–163.
  • K.A. Ross, Signed hypergroups - a survey, Applications of hypergroups and related measure algebras, 319–329, Connett, William C. (ed.) et al., American Mathematical Society, Providence, 1995.
  • G. Szegő, Orthogonal polynomials, American Mathematical Society, Providence, 1975.
  • R. Szwarc, Connection coefficients of orthogonal polynomials, Canad. Math. Bull. 35 (1992), 548–556.
  • W.F. Trench, Nonnegative and alternating expansions of one set of orthogonal polynomials in terms of another, SIAM J. Math. Anal. 4 (1973), 111–115.
  • M. Vogel, Spectral synthesis on algebras of orthogonal polynomial series, Math. Z. 194 (1987), 99–116.
  • M.W. Wilson, Nonnegative expansions of polynomials, Proc. Amer. Math. Soc. 24 (1970), 100–102.