Ground state representations of topological groups

Abstract

AbstractLet $$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)$$ α : R → Aut ( G ) define a continuous $${{\mathbb {R}}}$$ R -action on the topological group G. A unitary representation $$(\pi ^\flat ,\mathcal {H})$$ ( π ♭ , H ) of the extended group $$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}$$ G ♭ : = G ⋊ α R is called a ground state representation if the unitary one-parameter group $$\pi ^\flat (e,t) = e^{itH}$$ π ♭ ( e , t ) = e itH has a non-negative generator $$H \ge 0$$ H ≥ 0 and the subspace $$\mathcal {H}^0 := \ker H$$ H 0 : = ker H of ground states generates $$\mathcal {H}$$ H under G. In this paper, we introduce the class of strict ground state representations, where $$(\pi ^\flat ,\mathcal {H})$$ ( π ♭ , H ) and the representation of the subgroup $$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )$$ G 0 : = Fix ( α ) on $$\mathcal {H}^0$$ 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 $$G^0$$ G 0 . This is particularly effective if the occurring representations of $$G^0$$ 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.