Neeb, Karl-Hermann; Russo, Francesco G. Ground state representations of topological groups. (English) Zbl 07796278 Math. Ann. 388, No. 1, 615-674 (2024). Summary: Let \(\alpha : \mathbb{R} \rightarrow \mathrm{Aut}(G)\) define a continuous \(\mathbb{R}\)-action on the topological group \(G\). A unitary representation \((\pi^\flat ,\mathcal{H})\) of the extended group \(G^\flat := G \rtimes_\alpha \mathbb{R}\) is called a ground state representation if the unitary one-parameter group \(\pi^\flat (e,t) = e^{itH}\) has a non-negative generator \(H \ge 0\) and the subspace \(\mathcal{H}^0 := \ker H\) of ground states generates \(\mathcal{H}\) under \(G\). In this paper, we introduce the class of strict ground state representations, where \((\pi^\flat ,\mathcal{H})\) and the representation of the subgroup \(G^0 := \mathrm{Fix}(\alpha )\) on \(\mathcal{H}^0\) 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 \(G^0\). This is particularly effective if the occurring representations of \(G^0\) 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. MSC: 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E66 Analysis on and representations of infinite-dimensional Lie groups 43A75 Harmonic analysis on specific compact groups 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Arveson, W., On groups of automorphisms of operator algebras, J. Funct. Anal., 15, 217-243 (1974) · Zbl 0296.46064 [2] Banaszczyk, W., Additive Subgroups of Topological Vector Spaces (1991), Berlin: Springer, Berlin · Zbl 0743.46002 [3] Beltiţă, D.; Neeb, K-H, A non-smooth continuous unitary representation of a Banach-Lie group, J. Lie Theory, 18, 933-936 (2008) · Zbl 1203.22013 [4] Beltiţă, D.; Ratiu, TS; Tumpach, AB, The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, J. Funct. 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