Pull-back properties of moment functions and a generalization of Bochner-Weil's theorem
Let \((\mathbb{S}, +,*)\) be a topological Abelian *-semigroup, let \(\mathbb{S}^*\) be the set of semicharacters which are bounded with respect to a function \(\gamma\), called absolute value, and let \(\mathbb{S}^*_c\) denote the subset of continuous \(\gamma\)-bounded semicharacters. Define in the usual way for a nonnegative Radon measure \(\lambda\) on \(\mathbb{S}^*\) the generalized Laplace transform \(\widehat \lambda (s)\), \(s \in \mathbb{S}\). The aim of the paper under review is to prove for a large class of *-semigroups with identity a Bochner-Weil-theorem. These *-semigroups, called \(\gamma\)-admissible, generalize e.g. discrete semigroups, locally compact Abelian groups, the semigroup ([0,1], min). The main results are a characterization of \(\mathbb{S}_c^*\) within \(\mathbb{S}^*\) (Theorem A) and of generalized Laplace transforms (Theorem B). Theorems \(C-E\) contain Bochner-Weil theorems for interesting special cases of \(\gamma\)-admissible *-semigroups.
scientific article; zbMATH DE number 701008