On the existence of regular vectors
Christoph ZellnerFriedrich-Alexander-Universität Erlangen-Nürnberg, Germany
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Let be a locally convex Lie group and be a continuous unitary representation. is called smooth if the space of -smooth vectors is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra of , a continuous unitary representation of 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 is called semibounded, if is smooth and there exists a non-empty open subset such that the operators from the derived representation are uniformly bounded from above for .