Perfectness of certain subsemigroups of the perfect semigroup (Q1122005)

Let S be an abelian semigroup with involution \(s\mapsto s^*\) and identity 0. A function \(\rho\) : \(S\to {\mathbb{C}}\) is called a semicharacter if (i) \(\rho (0)=1\), (ii) \(\rho (s+t)=\rho (s)\rho (t)\), and (iii) \(\rho (s^*)=\overline{\rho (s)}\). S is said to be perfect if any positive definite function on S can be represented as an integral of semicharacters with unique measure. In this article, it is proved that if T is a *-subsemigroup of the perfect semigroup S and satisfies \(t+S\subset T\) for all \(t\in T\setminus \{0\}\) then T is also perfect. As application of this result, several examples of such *-subsemigroups are given.