Existence of proper semihypergroups of type U on the right

We generalize the classical definition of hypergroups of type U on the right to semihypergroups, and we prove some properties of their subsemihypergroups and subhypergroups. In particular, we obtain that a finite proper semihypergroup of type U on the right can exist only if its order is at least 6. We prove that one such semihypergroup of order 6 actually exists. Moreover, we show that there exists a hypergroup of type U on the right of cardinality 9 containing a proper non-trivial subsemihypergroup. In this way, we solve a problem left open in [D. Freni, Sur les hypergroupes de type U et sous-hypergroupes engendrés par un sous-ensemble, Riv. Mat. Univ. Parma 13 (1987) 29-41].     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Hypergroups and polygroups in diffeology

Abstract

We introduce diffeological algebraic hyperstructures such as diffeological hypergroups and diffeological polygroups, which are generalizations of diffeological groups. After providing some examples of these notions, we investigate the relationships between diffeological hypergroups and diffeological groups. It is proved that there is an equivalence of categories between subductions over diffeological groups and diffeological complete hypergroups. We finally show how every diffeological polygroup is a subpolygroup of a diffeological poly-monoid of hyper-diffeomorphisms.

Communicated by J. L. Gomez Pardo     

Central limit theorem and infinitely divisible probabilities associated with partial differential operators

We provide

SU(d)-Biinvariant Random Walks on SL\,{\left( {d,\,\mathbb{C}} \right)} and their Euclidean Counterparts

We establish a deformation isomorphism between the algebras of -biinvariant compactly supported measures on and -conjugation invariant measures on the Euclidean space of all Hermitian -matrices with trace . This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from on the other, which has recently been observed by Klyachko [13]. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded -biinvariant measures on and a certain (non-invariant) subalgebra of the bounded signed measures on . We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of -biinvariant random walks on in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.Margit Rösler was partially supported by the Netherlands Organisation for Scientific Research (NWO), project nr. B 61-544.     

Multipliers on Vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces for Hypergroups

In this article, we extend the well known Wendel’s theorem to the context of vector-valued L¹-spaces on hypergroups. In this regard, various cases have been studied.     

Actions of Finite Hypergroups

This paper is concerned with actions of finite hypergroups on sets. After introducing the definitions in the first section, we use the notion of maximal actions to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.     

The Semisimplicity of L^1 （K,w） of a Weighted Commutative Hypergroup K

In the present paper, it is shown that, for a locally compact commutative hypergroup K with a Borel measurable weight function w, the Banach algebra L^1 （K, w） is semisimple if and only if L^1 （K） is semisimple. Indeed, we have improved compact groups to the general setting of locally a well-krown result of Bhatt and Dedania from locally compact hypergroups.     

Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Let K be respectively the parabolic biangle and the triangle in and be a sequence in [0, +[ such that lim_p (p)=+. According to Koornwinder and Schwartz,⁽⁷⁾ for each there exist a convolution structure (*_(p)) such that (K, *_(p)) is a commutative hypergroup. Consider now a random walk on (K, *_(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.     

Hypoelliptic Jacobi Convolution Operators on Schwartz Distributions

In this paper we characterize the hypoellipticity of Jacobi convolution operators on Schwartz distributions. In the proof of the main result of this paper the positivity ofthe convolution structure for the inverse of the Jacobi transform plays an essential role. We also study hypoelliptic convolution equations on Chébli-Triméche hypergroups.Mathematics Subject Classification 2000: 46F12