Ground state representations of topological groups

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dc.contributor.authorNeeb, Karl-Hermann
dc.contributor.authorRusso, Francesco G
dc.date.accessioned2025-01-17T10:00:39Z
dc.date.available2025-01-17T10:00:39Z
dc.date.issued2024
dc.description.abstractLet α : R → Aut(G) define a continuous R-action on the topological group G. A unitary representation (π , H) of the extended group G := G α R is called a ground state representation if the unitary one-parameter group π (e, t) = eitH has a nonnegative generator H ≥ 0 and the subspace H0 := ker H of ground states generates H under G. In this paper, we introduce the class of strict ground state representations, where (π , H) and the representation of the subgroup G0 := Fix(α) on H0 have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of G0. This is particularly effective if the occurring representations of G0 can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations
dc.identifier.citationNeeb, K.H. and Russo, F.G., 2024. Ground state representations of topological groups. Mathematische Annalen, 388(1), pp.615-674.
dc.identifier.urihttps://doi.org/10.1007/s00208-022-02531-4
dc.identifier.urihttps://hdl.handle.net/10566/19815
dc.language.isoen
dc.publisherSpringer Science and Business Media Deutschland GmbH
dc.subjectGround state representations
dc.subjectTopological groups
dc.subjectClassification
dc.subjectUnitary representations
dc.subjectLie groups
dc.titleGround state representations of topological groups
dc.typeArticle

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