A generalization of Bochner-Weil?s theorem and Stone?s theorem on foundation semigroups

Abstract

In this paper, we improve a result of Youssfi. As a consequence we obtain an extension for the Bochner-Weil theorem and the Stone theorem on a weighted foundation * -semigroup with an identity whose weight is assumed to be bounded on a neighbourhood of the identity. Introduction.In 1994, Youssfi in his paper [11] gave a positive answer to a problem raised by Choquet in [3] by proving the Bochner-Weil theorem for the w-bounded and continuous at the identity positive definite functions on commutative * -semigroups with an identity and with a weight function w continuous at the identity. Recently, Ressel and Ricker proved a version of the Stone theorem on commutative discrete semigroups (see, [8]).In the present paper, we shall first extend Youssfi's result to the case of topological * -semigroups with weight functions which are bounded only on a neighbourhood of the identity. Applications of the obtained results has provided us with a generalization of the Bochner-Weil theorem on foundation * -semigroups and topological * -groups with weight functions not necessary bounded on a neighbourhood of the identity. We have closed the paper with a generalization of the Ressel and Ricker version of the Stone theorem on foundation * -semigroups. It should be noted that the family of foundation * -semigroups is very large, for which discrete * -semigroups and locally compact * -groups are elementary examples. For a wide class of examples of such semigroups we refer the interested reader to the Appendix B of [9].