POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS
Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 295-316
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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♯ := G ⋊αT 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♯.
NEEB, KARL-HERMANN. POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS. Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 295-316. doi: 10.1017/S0017089513000268
@article{10_1017_S0017089513000268,
author = {NEEB, KARL-HERMANN},
title = {POSITIVE {ENERGY} {REPRESENTATIONS} {AND} {CONTINUITY} {OF} {PROJECTIVE} {REPRESENTATIONS} {FOR} {GENERAL} {TOPOLOGICAL} {GROUPS}},
journal = {Glasgow mathematical journal},
pages = {295--316},
year = {2014},
volume = {56},
number = {2},
doi = {10.1017/S0017089513000268},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000268/}
}
TY - JOUR AU - NEEB, KARL-HERMANN TI - POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS JO - Glasgow mathematical journal PY - 2014 SP - 295 EP - 316 VL - 56 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000268/ DO - 10.1017/S0017089513000268 ID - 10_1017_S0017089513000268 ER -
%0 Journal Article %A NEEB, KARL-HERMANN %T POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS %J Glasgow mathematical journal %D 2014 %P 295-316 %V 56 %N 2 %U http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000268/ %R 10.1017/S0017089513000268 %F 10_1017_S0017089513000268
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