Perfectness and semiperfectness of abelian $*$-semigroups without zero

Stieltjes Perfect Semigroups are Perfect

“…A complex-valued function on a * -semigroup S is Stieltjes singular if it vanishes on S + S. For s ∈ S define G s = {σ ∈ S * : σ(s) = 0} and G If S is a perfect semigroup then so isS [25]. If S is a * -semigroup having a zero 0 then σ(0) = 1 for all σ ∈ S * , as is readily seen.…”

“…. Since the paper is already quite long, we merely refer to the corresponding result for perfectness instead of Stieltjes perfectness [25].…”

A Note on Factoring of Positive Definite Functions on Semigroups

“…Existence: Let S = H ∪ {0} be the semigroup with zero obtained by adjoining to H a zero external to H. From the fact that H is quasi-perfect it follows that S is perfect (see [6]). The following facts are taken from [8], proof of Theorem 3.2 (where the measures were assumed to be Radon measures but the proof carries over). For y ∈ H + H there is a unique complex measure µ y on A(S * ) such that…”

“…for all x ∈ S. (Given y ∈ H + H, choose h, k ∈ H such that y = h + k * and let µ y be the measure denoted by µ h,k in [8] where ϕ s (y) = µ y 1 {0} and ϕ m (y) = G σ(y) dµ(σ). In general, however, the set 1 {0} need not be measurable.…”

A reduction of the problem of characterizing perfect semigroups

“…If X is perfect then H is quasi-perfect by Theorem 2.4 since H is an ideal of X. Conversely, if H is quasi-perfect then the perfectness of X follows just as in the proof of [6], Theorem 3.2, that perfectness of H implies perfectness of H ∪ {0}, only the argument with the Cauchy-Schwarz inequality has to be applied twice, first to get from the integral representation on H + H + H + H (⊂ H + H + H ) to the integral representation on H + H and a second time to get it on all of H . Corollary 2.6.…”