On the existence of regular vectors

  • Christoph Zellner

    Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
On the existence of regular vectors cover
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Let GG be a locally convex Lie group and π:GU(H)\pi:G \to \mathrm{U}(\mathcal H) be a continuous unitary representation. π\pi is called smooth if the space of π\pi-smooth vectors HH\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 g\mathfrak{g} of GG, a continuous unitary representation of GG 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\pi is called semibounded, if π\pi is smooth and there exists a non-empty open subset UgU\subset\mathfrak{g} such that the operators id pi(x)i \mathrm d\ pi(x) from the derived representation are uniformly bounded from above for xUx \in U.