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Journal of Lie Theory 18 (2008), No. 4, 933--936
Copyright 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

Karl-Hermann Neeb
Dept. of Mathematics, University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany


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

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