On perfect subsemigroups of \(\mathbb Q_+\) (Q1603253)

Let \((S,+,\ast)\) be an abelian semigroup with involution, and \(\mathcal P\) and \(S^\ast\) the sets of positive definite functions on \(S\) and of characters, respectively. \(S\) is said to be perfect if for every \(\varphi\in\mathcal P\) there is a unique measure \(\mu\) on \(\mathcal A\), the smallest \(\sigma\)-algebra on \(S^\ast\) rendering the functions \(\sigma\to\sigma(s)\) \(S^\ast\to \mathbb{C}\), \(s\in S\), measurable such that \(\varphi(s)=\int\sigma(s) d\mu(\sigma)\) for \(s\in S\). For every subset \(V\) of \(S\), denote by \(E(V)\) the set of those \(v\in V\) such that the conditions \(s,t\in S\), \(s+s^\ast,t+t^\ast\in V\), and \(s+t^\ast=v\) imply \(s=t\). For every subset \(U\) of \(S\), denote by \(C(U)\) the union of all finite subsets \(V\) of \(S\) such that \(E(V)\subset U\). The semigroup \(S\) is \(C\)-finite if \(C(U)\) is a finite set for every finite subset \(U\) of \(S\). In a recent paper the author [Acta Math. Hung. 79, No. 4, 269-294 (1998; Zbl 0909.20047)] proved that in order for a subsemigroup \(S\) of a perfect semigroup \({\mathbb{Q}_+}\) properly containing \(\{0\}\) to be perfect it is necessary that \(S\) be non-\(C\)-finite. In the present paper he shows that there is a non-\(C\)-finite subsemigroup of \({\mathbb{Q}_+}\) which is not perfect. The necessity of the strictly stronger condition of non-quasi-\(C\)-finiteness is also proved.