Positive semicharacters of Lie semigroups

Abstract

We study positive semicharacters of generating Lie subsemigroup ,l 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 Sf consisting of all bounded positive semicharacters of S. In particular Sf is nontrivial for a solvable iimply connected G and invariant ,S without nontrivial subgroups, but it is trivial for a semisimple G.