Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 18 (2008), No. 4, 933--936Copyright Heldermann Verlag 2008 A Nonsmooth Continuous Unitary Representation of a Banach-Lie Group Daniel Beltita Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, 70700 Bucharest, Romania Daniel.Beltita@imar.ro Karl-Hermann Neeb Dept. of Mathematics, University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany neeb@mathematik.tu-darmstadt.de [Abstract-pdf] We show that the representation of the additive group of the Hilbert space $L^2([0,1],{\mathbb R})$ on $L^2([0,1], {\mathbb C})$ given by the multiplication operators $\pi(f) := e^{if}$ is continuous but its space of smooth vectors is trivial. This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth. Keywords: Infinite-dimensional Lie group, unitary representation, smooth vector. MSC: 22E65, 22E45 [ Fulltext-pdf  (137  KB)] for subscribers only.