Stieltjes Perfect Semigroups are Perfect

An abelian *-semigroup S is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on S 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 S is perfect if for each s ∈ S there exist t ∈ S and m, n ∈ ℕ₀ such that m + n ≥ 2 and s + s* = s* + mt + nt*. This was known only with s = mt + nt* instead. The equality cannot be replaced by s + s* + s = s + s* + mt + nt* in general, but for semigroups with neutral element it can be replaced by s + p(s + s*) = p(s + s*) + mt + nt* for arbitrary p ∈ ℕ (allowed to depend on s).