Orthogonal Polynomials and Point and Weak Amenability of $$\ell ^1$$ ℓ 1 -Algebras of Polynomial Hypergroups

Abstract

In contrast to the group case, amenability and many of its various generalizations are rather strong conditions on the \(\ell ^1\)-algebra of a polynomial hypergroup. In this paper, we study weak amenability and investigate the nonexistence of nonzero bounded point derivations w.r.t. symmetric characters; originally coming from cohomology theory, here these notions correspond to rather concrete problems concerning derivatives of orthogonal polynomials. We give some general results and show that there are polynomial hypergroups with weakly amenable, but nonamenable \(\ell ^1\)-algebra—the latter answers a question which has been open for some years. Moreover, we characterize both point and weak amenability for the classes of ultraspherical, Jacobi, symmetric Pollaczek, random walk, associated ultraspherical and cosh-polynomials by identifying the corresponding parameter regions. Our methods are from the theory of orthogonal polynomials and special functions, and from harmonic and functional analysis; in particular, we shall use approximations, expansions, and asymptotics.