A L´evy]Khinchine Formula for Semigroups and Related Problems: An Adapted Spaces Approach

In this paper we use a Choquet type theorem on adapted spaces to obtain or reobtain some results in harmonic analysis on semigroups. Thus we give a L´evy]Khinchine formula for some negative definite functions defined on a commutative semigroup with neutral element, we prove that completely monotonic resp. alternating.functions are completely positive resp. negative. definite, we characterize the completely monotonic and the completely alternating func- tions defined on N*[ 1, 2, 3, . . .4, and we consider a Stieltjes’ moment problem.