# On the existence of regular vectors

### Christoph Zellner

Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

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## Abstract

Let $G$ be a locally convex Lie group and $π:G→U(H)$ be a continuous unitary representation. $π$ is called smooth if the space of $π$-smooth vectors $H_{∞}⊂H$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $g$ 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 $π$ is called semibounded, if $π$ is smooth and there exists a non-empty open subset $U⊂g$ such that the operators $idpi(x)$ from the derived representation are uniformly bounded from above for $x∈U$.