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

  • KARL-HERMANN NEEB (a1)

Abstract

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.

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References

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

  • KARL-HERMANN NEEB (a1)

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