Central measures on semisimple Lie groups have essentially compact support

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra ${L_1}(H)$ is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image.