Probability Theory on Semihypergroups

Motivated by the work of Hognas and Mukherjea on semigroups,we study semihypergroups, which are structures closer to semigroups than hypergroups in the sense that they do not require an identity or an involution. Dunkl[Du73] calls them hypergroups (without involution), and Jewett[Je75] calls them semiconvos. A semihypergroup does not assume any algebraic operation on itself. To generalize results from semigroups to semihypergroups, we first put together the fundamental algebraic concept a semihypergroup inherits from its measure algebra. Among other things, we define the Rees convolution product, and prove that if X; Y are non-empty sets and H is a hypergroup, then with the Rees convolution product, X x H x Y is a completely simple semihypergroup which has all its idempotent elements in its center. We also point out striking differences between semigroups and semihypergroups. For instance, we construct an example of a commutative simple semihypergroup, which is not completely simple. In a commutative semihypergroup S, we solve the Choquet equation μ * v = v, under certain mild conditions.We also give the most general result for the non-commutative case.We give an example of an idempotent measure on a commutative semihypergroup whose support does not contain an idempotent element and so could not be completely simple. This is in contrast with the context of semigroups, where idempotent measures have completely simple supports. The results of Hognas and Mukherjea [HM95] on the weak convergence of the sequence of averages of convolution powers of probability measures is generalized to semihypergroups. We use these results to give an alternative method of solving the Choquet equation on hypergroups (which was initially solved in [BH95] with many steps). We show that If S is a compact semihypergroup and μ is a probability measure with S = [ U∞n=1 Supp(μ)n], then for any open set G ⊃ K where K is the kernel of S limn-→∞μn(G) = 1. Finally, we extend to hypergroups basic techniques on multipliers set forth for groups in [HR70], namely propositions 5.2.1 and 5.2.2 , we give a proof of an extended version of Wendel\u27s theorem for locally compact commutative hypergroups and show that this version also holds for compact non-commutative hypergroups. For a compact commutative hypergroup H, we establish relationships between semigroup S = S = {T(ξ) : ξ \u3e 0} of operators on Lp(H), 1 ≤ p \u3c 1 \u3c ∞, which commutes with translations, and semigroup M = {Eξ : ξ \u3e 0} of Lp(H) multipliers. These results generalize those of [HP57] for the circle groups and [B074] for compact abelian groups