On the existence of regular vectors

Abstract

Let $G$ be a locally convex Lie group and $\pi :G\to \mathrm{U}(\mathcal{H})$ be a continuous unitary representation. $\pi$ is called smooth if the space of $\pi$-smooth vectors ${\mathcal{H}}^{\infty }\subset \mathcal{H}$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $\mathfrak{g}$ of $G$, a continuous unitary representation of $G$ is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach–Lie groups. Here $\pi$ is called semibounded, if $\pi$ is smooth and there exists a non-empty open subset $U\subset \mathfrak{g}$ such that the operators $i\mathrm{d}pi(x)$ from the derived representation are uniformly bounded from above for $x\in U$.