Analyticity and Injectivity of Convolution Semigroups on Lie Groups

We prove that all continuous convolution semigroups of probability distributions on an arbitrary Lie group are injective. Let {μt, t>0} be a continuous convolution semigroup of probability distributions on a Lie group G. For each t>0, we set Ttf(x)=∫G f(xy) μt(dy) for a bounded continuous function f. We show that Ttf=0 holds if and only if f=0. This fact will be applied in proving the unique divisibleness of the convolution product for a certain distribution. We show that ν * ξ=ν * ξ′ implies ξ=ξ′, provided that ν is an infinitely divisible distribution on a simply connected nilpotent Lie group.