Moment problems on subsemigroups of \({\mathbb{N}{}}^ k_ 0\) and \({\mathbb{Z}{}}^ k\)
Let \(S\) be an abelian semigroup with involution \(*\), and let \(S^*\) be its set of \(*\)- semicharacters (non-zero \(*\)-homomorphisms of \(S\) into \(\mathbb{C}\)). We call \(S\) semiperfect if every positive definite function \(\phi\) on \(S\) can be represented by the form \(\phi(s)=\int_{S^*}\chi(s)d\mu(\chi)\) where \(\mu\) is a bounded Radon measure on \(S^*\); further \(S\) is said to be perfect if the representing measure \(\mu\) is unique. In this paper it is shown that all non-trivial semiperfect subsemigroups of \(\mathbb{Z}^ k\) (with the identity involution) are singly generated and are never perfect.
scientific article; zbMATH DE number 64386