Positive Semicharacters of Lie Semigroups

Abstract

We study positive semicharacters of generating Lie subsemigroup $$S$$ of a connected Lie group $$G$$ . These semicharacters are important for positive representations of $$S$$ in Hilbert space and for completely monotonic functions in $$S$$ . We describe the tangent map for a positive semicharacter and then obtain a necessary and sufficient condition for nontriviality of the wedge $$S_1^*$$ consisting of all bounded positive semicharacters of $$S$$ . In particular $$S_1^*$$ is nontrivial for a solvable simply connected $$G$$ and invariant $$S$$ without nontrivial subgroups, but it is trivial for a semisimple $$G$$ .