Abstract. Let K be a commutative hypergroup with the Haar measure . In the present paper we investigate whether the maximal ideals in L 1 ðK; Þ have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L 1 ðK; Þ. Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.

Abstract. The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on No . If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monk recursion formula and random walk polynomials. Polynomial hypergroupsIn [18] we demonstrated a close relationship between certain hypergroups on No and certain orthogonal polynomial sequences. In this paper we discuss other basic properties concerning these hypergroups-we call them polynomial hypergroups-and give many examples based on certain classes of orthogonal polynomial sequences. Our main reference is [18]. Notation and results from [18] are used throughout.The best known class corresponding to a polynomial hypergroup are the Jacobi polynomials (PJT 'ß) (x))f?=Q fora>jff>-l,a + jff-r-l>0 (see [18, 3(a)]). The basic property (P) holds as is shown by Gasper [11]. In [12]Gasper also proved that (Pna \x))f¡Lo defines a dual hypergroup structure on the interval [-1, 1]. Recently, Connett and Schwartz [9] proved that the Jacobi polynomials with a > ß > -1 and either ß > -1/2 or a + ^ > 0 are the only ones which define a dual hypergroup structure on [- (see e.g. [20-23, 32-34]). We reveal further polynomial hypergroups and determine their recurrence relations, their Haar weights and specific properties of the corresponding polynomials. In the following we consider the case of even orthogonalization measures only. Also, we choose x = 1 as the point

Abstract. For a locally compact group G and a group B of topological automorphisms containing the inner automorphisms of G and being relatively compact with respect to Birkhofftopology (that is G e [FIA]j~, B ~_ I (G)) the space Gz~ of B-orbits is a commutative hypergroup (= commutative convo in JEWETT's terminology) in a natural way as JEWETT has shown. Identifying the space of hypergroup characters of G z with E (G,B) (the extreme points of B-invariant positive definite continuous functions p with p (e) = l, endowed with the topology of compact convergence) we prove that E (G, B) is a hypergroup, the "hypergroup dual" of G z. Introduction The theory of topological hypergroups was initiated by DUXKL 16'.'

No abstract

Dedicated to Professor Elmar Thoma on the occasion of his 60 th birthdayThe purpose of this paper is to establish a unified treatment of many disparate theorems of Levy-Hinέin type. The appropriate framework to do this is the theory of commutative hypergroups. In this way we not only generalize the results mentioned above but also settle some asymmetries indicated above. Roughly speaking a hypergroup K is a space in which the product of two elements is a probability measure on this space satisfying certain conditions. If K is commutative and if the space K of characters is a hypergroup under pointwise operations a Levy-Hincin formula for convolution semigroups is obtained. Before setting up some notation we show how the examples fit in.