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POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS

Published online by Cambridge University Press:  13 August 2013

KARL-HERMANN NEEB*
Affiliation:
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany e-mail: neeb@math.fau.de
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Abstract

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Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G := GαT the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π, ${\mathcal{H}}$) of G whose restriction to G remains irreducible. First, we prove that, for T = ${\mathbb R}$, this is the case for any irreducible positive energy representation of G, i.e. for which the one-parameter group Ut := π(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

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