Convergence of Semigroups of Complex Measures on a Lie Group

A theorem of Siebert in its essential part asserts that if μ n (t) are semigroups of probability measures on a Lie group G, and P n are the corresponding generating functionals, then $$\bigl \langle \mu_n(t),f \bigr \rangle \ \xrightarrow[n]{}\ \bigl \langle \mu_0(t),f \bigr \rangle , \quad f\in C_b(G), \ t>0,$$ implies $$\langle \pi_{P_n}u,v\rangle \ \xrightarrow[n]{}\ \langle \pi_{P_0}u,v\rangle ,\quad u\in C^{\infty}(E,\pi), \ v\in E,$$ for every unitary representation π of G on a Hilbert space E, where C ∞(E,π) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements, the most important being the extension of the theorem to semigroups of complex measures. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis $$\langle P_n,f\rangle \ \xrightarrow[n]{}\ \langle P_0,f\rangle ,$$ for bounded twice differentiable functions f. As a corollary, the above thesis of Siebert is extended to bounded strongly continuous representations of G on Banach spaces.