Let S be an abelian *-semigroup. In this paper we prove some equivalent conditions for that every positive function is a moment function on S + S + S with a unique representation measure on the set of characters of S.
An abelian -semigroup is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian -semigroup is perfect if for each there exist and such that ...
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