Class Of Hypergroups Research Articles

Complete parts and subhypergroups in reversible regular hypergroups

Abstract In this paper we analyse the center and centralizer of an element in the context of reversible regular hypergroups, in order to obtain the class equation in regular reversible hypergroups, by using complete parts. After an introduction in which basic notions and results of hypergroup theory are presented, particularly complete parts, then we give several properties, characterisations and also examples for the center and centralizer of an element for two classes of hypergroups. The next paragraph is dedicated to hypergroups associated with binary relations. We establish a connection between several types of equivalence relations, introduced by J. Jantosciak, such as the operational relation, the inseparability and the essential indistin-guishability and the conjugacy relation for complete hypergroups. Finally, we analyse Rosenberg hypergroup associated with a conjugacy relation.

On the fourier algebra of certain hypergroups

We consider certain classes of hypergroups and address some problems regarding their Fourier algebras, including zero product preserving maps, local derivations and weak amenability. The key tool in our approach is to show that the Fourier algebra of such hypergroups has the so called property (𝔹). This property has already been shown to be very effective in solving problems related to zero product preserving maps and local derivations provided that some other conditions are satisfied. Some examples of groups and hypergroups, the Fourier algebras of which have the property (𝔹) satisfying all those conditions are given. Our examples are the class of almost abelian groups and double coset hypergroups of Gelfand pairs. We also prove that if H is the double coset hypergroup of a Gelfand pair, then A(H) is weakly amenable. We provide some results on the direct product of commutative (regular) Fourier hypergroups.

On some classes of hypergroups

In this paper, we first introduce the notion of power(ful) hypergroups, proving that the classes of cancelative semigroups and complete hypergroups are powerful hypergroups. We investigate some pro...

Characters of induced representations of a compact hypergroup

Let \(K\) be a compact hypergroup and \(K_0\) a closed subhypergroup of \(K\). For a state \(\varphi \) of the Banach \(*\)-algebra \(M^b(K_0)\) the induced state \(\mathrm{ind}_{K_0}^K \varphi \) of \(M^b(K)\) is introduced. Applying the notion of a character mapping \(ch\) from the space \(\mathrm{Rep}^\mathrm{f}(K)\) of finite-dimensional representations of \(K\) into the space \(M^1(\hat{K})\) of probability measures on the space \(\hat{K}\) of equivalence classes of irreducible representations of \(K\). For certain classes of hypergroups \(K\) it is shown that, whenever \(\pi \in \mathrm{Rep}^\mathrm{f}(K_0)\), the formula \(ch(\mathrm{ind}_{K_0}^K\pi ) = \mathrm{ind}_{K_0}^Kch(\pi )\) holds.

Configurations and invariant nets for amenable hypergroups and related algebras

Let H H be a hypergroup with left Haar measure. The amenability of H H can be characterized by the existence of nets of positive, norm one functions in L 1 ( H ) L^1(H) which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of H H using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in L 1 ( H ) L^1(H) which tend to left invariance weakly, but not in norm. We define the semidirect product of H H with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product. These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra.

Compact Hypergroup Characters and Finite Universal Korovkin Sets in L 1-Hypergroup Algebra

Let K be a compact hypergroup.We investigate Trig(K), the linear span of coordinate functions of the irreducible representations of K. Contrary to the group case, Trig(K) endowed with the usual multiplication does not bear an algebra structure, but it has a natural normed algebra structure when it inherits the convolution from $$\mathcal {M}(K)$$ , the algebra of all bounded Radon measures. We characterize the center of the algebras $$\mathcal {M}(K)$$ , L p (K) and Trig(K) respectively, and consequently we obtain, for a certain class of hypergroups, the correspondence between the structure space of the center of L 1(K) and the center of Trig(K). As an application we study the existence of a finite universal Korovkin set w.r.t. positive operators in the center of L 1(K), in particular in L 1(K), whenever K is commutative.

One-dimensional hypergroups

It is a well-known fact that on the real line R and on the torus % there exists-up to isomorphy-exactly one group operation which is compatible with the (usual) topology on these spaces. However, on the half-line Iw + and on the compact interval [0, l] no structure of a topological group exists. If we consider instead of groups the convolution structure of hypergroups (see [ 12, 15, 11 I), the situation becomes completely different. On R + and [0, l] there exist large classes of nonisomorphic hypergroups. On the other hand, on R and U no further hypergroups can be defined. It is the aim of this article to investigate the properties of hypergroups on the one-dimensional spaces 03, T, R + , and [0, l] which can be deduced from the topological and geometrical structure of these spaces. In the following section, elementary facts about one-dimensional hypergroups concerning the neutral element and the involution will be proved. As a consequence, each of these hypergroups is commutative. It will be shown in the third section that the extreme points of the support of two point masses can be calculated easily. In Section 4 the classes of hypergroups on R + and [0, l] introduced by Chebli [4] and Achour and Trim&he [l, 17) will be considered. Homomorphisms between hypergroups and the relation to stable measures are investigated in Section 5. In the last section we enumerate those hypergroups which are double coset spaces of locally compact groups.