Analytic Extension Techniques for Unitary Representations of Banach–Lie Groups

Merigon, Stéphane; Neeb, Karl-Hermann

doi:10.1093/imrn/rnr174

Cited by 11 publications

(16 citation statements)

References 20 publications

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“…This generalizes a result in [10] where it is assumed that an Olshanski semigroup G exp(iW π ) ⊂ G C in a complexification G C of G exists, cf. [10,Def. 5.3 & Thm.…”

Section: Introductionsupporting

confidence: 75%

On the existence of regular vectors

Zellner

2017

EMS Series of Congress Reports

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Let G be a locally convex Lie group and π : G → U(H) be a continuous unitary representation. π is called smooth if the space of π-smooth vectors H ∞ ⊂ H is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra g of G, a continuous unitary representation of G is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach-Lie groups. Here π is called semibounded, if π is smooth and there exists a non-empty open subset U ⊂ g such that the operators idπ(x) from the derived representation are uniformly bounded from above for x ∈ U .

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“…This generalizes a result in [10] where it is assumed that an Olshanski semigroup G exp(iW π ) ⊂ G C in a complexification G C of G exists, cf. [10,Def. 5.3 & Thm.…”

Section: Introductionsupporting

confidence: 75%

On the existence of regular vectors

Zellner

2017

EMS Series of Congress Reports

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“…[LM75], [HN93,Sect. 9.5] and [MN11] for a generalization to Banach-Lie groups). We therefore focus on the triple (G, τ, S) and unitary representations as above.…”

Section: Introductionmentioning

confidence: 99%

Reflection positivity and conformal symmetry

Neeb

Ólafsson

2014

Journal of Functional Analysis

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A reflection positive Hilbert space is a triple (E , E+, θ), where E is a Hilbert space, θ a unitary involution and E+ a closed subspace on which the hermitian form v, w θ := θv, w is positive semidefinite. From this data one obtains a Hilbert space E by completing a suitable quotient of E+ with respect to ·, · θ on E+. To obtain compatible unitary representations of Lie groups, we start with triples (G, S, τ ), where G is a Lie group, τ an involutive automorphism of G and S a subsemigroup invariant under the involution) and π(S)E+ ⊆ E+. Motivated by the passage from the euclidean motion group to the Poincaré group in quantum field theory, one expects a duality between reflection positive representations and unitary representations of the dual symmetric Lie group G c on E. We propose a new approach to a reflection positive representations based on reflection positive distributions and reflection positive distribution vectors. In particular, we generalize the Bochner-Schwartz Theorem to positive definite distributions on open convex cones and apply our techniques to complementary series representations of the conformal group O + 1,n+1 (R) of the sphere S n .

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“…As Olshanski semigroups and the extension of unitary representations also works to some extent for Banach-Lie groups [MN12], one may expect that large portions of our results can be generalized to Banach-Lie groups endowed with a suitably continuous action of R × , encoding the modular objects. 6.3.2.…”

Section: Wick Rotations For Non-uniformly Continuous Actionsmentioning

confidence: 83%

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Neeb¹

2021

Represent. Theory

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Let V \mathtt {V} be a standard subspace in the complex Hilbert space H \mathcal {H} and G G be a finite dimensional Lie group of unitary and antiunitary operators on H \mathcal {H} containing the modular group ( Δ V i t ) t ∈ R (\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} of V \mathtt {V} and the corresponding modular conjugation J V J_{\mathtt {V}} . We study the semigroup \[ S V = { g ∈ G ∩ U ⁡ ( H ) : g V ⊆ V } S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge L ⁡ ( S V ) = { x ∈ g : exp ⁡ ( R + x ) ⊆ S V } \operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\} , i.e., the generators of its one-parameter subsemigroups in the Lie algebra g \mathfrak {g} of G G . The semigroup S V S_{\mathtt {V}} is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp ⁡ ( i C ) G \exp (iC) , where C ⊆ g C \subseteq \mathfrak {g} is an Ad ⁡ ( G ) \operatorname {Ad}(G) -invariant closed convex cone. Our main results assert that the Lie wedge L ⁡ ( S V ) \operatorname {\textbf {L}}(S_{\mathtt {V}}) spans a 3 3 -graded Lie subalgebra in which it can be described explicitly in terms of the involution τ \tau of g \mathfrak {g} induced by J V J_{\mathtt {V}} , the generator h ∈ g τ h \in \mathfrak {g}^\tau of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup S V S_{\mathtt {V}} itself.

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Analytic Extension Techniques for Unitary Representations of Banach–Lie Groups

Cited by 11 publications

References 20 publications

On the existence of regular vectors

On the existence of regular vectors

Reflection positivity and conformal symmetry

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

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