1Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, 70700 Bucharest, Romania 2Dept. of Mathematics, University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Journal of Lie Theory, Tome 18 (2008) no. 4, pp. 933-936
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.
1
Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, 70700 Bucharest, Romania
2
Dept. of Mathematics, University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Daniel Beltita; Karl-Hermann Neeb. A Nonsmooth Continuous Unitary Representation of a Banach-Lie Group. Journal of Lie Theory, Tome 18 (2008) no. 4, pp. 933-936. http://geodesic.mathdoc.fr/item/JOLT_2008_18_4_a11/
@article{JOLT_2008_18_4_a11,
author = {Daniel Beltita and Karl-Hermann Neeb},
title = {A {Nonsmooth} {Continuous} {Unitary} {Representation} of a {Banach-Lie} {Group}},
journal = {Journal of Lie Theory},
pages = {933--936},
year = {2008},
volume = {18},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2008_18_4_a11/}
}
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