On an invariance property of the space of smooth vectors

Russo

2022

Math. Ann.

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Let $$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)$$ 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.

“…Definition A.2 (cf. [33,40]) Let V be a complete complex locally convex space and let α : R → GL(V ), t → α t be a strongly continuous representation.…”

Section: Ground Statementioning

confidence: 99%

“…Theorem A.12 (Spectral translation formula; [40,Thm. 3.1]) Assume that g is a complete locally convex Lie algebra, α : R → Aut(G) defines a continuous action of R on G, and that the induced action on g C is also continuous.…”

Section: Ground Statementioning

confidence: 99%

Ground state representations of topological groups

Russo

2022

Math. Ann.

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“…The reproducing kernel of H , as defined in (19), is given byK(g, g ) = P W π(g) −1 π(g )P * W , and therefore it satisfies (22). …”

Section: Gl(w )mentioning

confidence: 99%

Classification of Positive Energy Representations of the Virasoro Group

Salmasian

2014

Int Math Res Notices

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“…This then leads to the study of smoothing operators (cf. Theorem 5.9 below, or [NSZ15]) which provide a method for constructing host algebras for some infinite dimensional Lie groups. In particular, the structure of these host algebras makes it easier to construct crossed product hosts for the groups involved.…”

Section: Covariant Representations Of Actions On Topological Groupsmentioning

confidence: 99%

“…We already considered the cross property for a few of these systems (cf. Examples 6.11 and 9.1-9.3 in [GrN14]), but here we want to pursue the general analysis of this question, because new mathematical tools such as smoothing operators for infinite dimensional Lie groups have recently become available ( [NSZ15]).…”

Section: Introductionmentioning

confidence: 99%

Crossed products ofC⁎-algebras for singular actions

Grundling

Journal of Functional Analysis

2014

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We analyze existence of crossed product constructions of Lie group actions on C *algebras which are singular. These are actions where the group need not be locally compact, or the action need not be strongly continuous. In particular, we consider the case where spectrum conditions are required for the implementing unitary group in covariant representations of such actions. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, we prove that the existence of cross representations is stable with respect to a large set of perturbations of the action, and we fully analyze the structure of cross representations of inner actions on von Neumann algebras. For one-parameter automorphism groups we study the cross property for covariant representations, where the generator of the implementing unitary group is positive. In particular, we find that if the Borchers-Arveson minimal implementing group is cross, then so are all other implementing groups. We study a smoothing phenomenon for one-parameter actions on Lie groups, and display the usefulness of cross representations for this context. For higher dimensional Lie group actions, we consider a class of spectral conditions which include the ones occurring in physics, and is sensible also for non-abelian or for infinite dimensional Lie groups. We prove that the cross property of a covariant representation is fully determined by the cross property of a certain one-parameter subsystem. This greatly simplifies the analysis of the existence of cross representations, and it allows us to prove the cross property for several examples of interest to physics. We also consider non-abelian extensions of the Borchers-Arveson theorem. There is a full extension in the presence of a cyclic invariant vector, but otherwise one needs to determine the vanishing of lifting obstructions.