Separation by characters or positive definite functions
Let \(S\) be a commutative semigroup with neutral element and (perhaps) with involution. It is well known that in general not every positive definite function on \(S\) is a mixture of characters (already on \(\mathbb{N}_0 \times \mathbb{N}_0\) there are counterexamples). Nevertheless, the author proves that if \(a,b\in S\) and \(\sigma(a)=\sigma(b)\) for each character \(\sigma\), then necessarily \(\varphi(a)= \varphi (b)\) for each positive definite function. The proof is highly nontrivial and shows deep insight into the fascinating area of ``Harmonic analysis on semigroups''.
scientific article; zbMATH DE number 943671