Positive semicharacters of Lie semigroups
A subsemigroup \(S\) of a Lie group \(G\) is called a Lie semigroup if it is the closure of a semigroup generated by one-parameter subsemigroups. In the paper under review the author studies bounded semicharacters of \(S\), i.e. homomorphisms into the multiplicative semigroup \(([0,1],\cdot)\). Using results of Devinatz and Nussbaum, he obtains integral representations of completely monotonic functions on \(S\) and spectral measures representing representations by positive operators. In Section 3 he deals with the relation between bounded semicharacters and their differentials. I have the impression that this section would have gained if the concept of the free group \(G(S)\) on a Lie semigroup would have been used [see \textit{J. Hilgert} and \textit{K.-H. Neeb}, ``Lie semigroups and their applications'', Lect. Notes Math. 1552 (Berlin 1993; Zbl 0807.22001)]. Since \(G(S)\) is a covering group of the group \(G\) containing \(S\), the universal property of \(G(S)\) provides for each semicharacter \(\chi\) a Lie group homomorphism \(G(S) \to ( {\text R}^\times, \cdot)\) and therefore a differential. In Section 4 the existence of non-trivial semicharacters is discussed. In particular it is shown that if \(G\) is simply connected solvable, \(S \cap S^{-1} = \{e\}\) and \(S\) is invariant, then non-trivial bounded semicharacters exist.
scientific article; zbMATH DE number 1278715