Perfectness of conelike * - Semigroups in double-struck Q sign^k

Let S be an abelian * - semigroup in double-struck Q sign^k. We give a sufficient condition for every positive definite function on S to have a unique representing measure on the dual semigroup of 5 (i. e. S is perfect). To characterize perfectness for any abelian * - semigroup is a challenging, but not yet generally solved problem. In this paper, we characterize the structure of involutions on an abelian * - semigroup which is a subset of double-struck Q sign^k, and show that any conelike * - semigroups in double-struck Q sign^k are perfect.