On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 1839-1874

Let G be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra 𝔤 we define the concept of an analytic functional and show that every positive analytic functional λ is integrable in the sense that it is of the form λ(D)=dπ(D)v,v for an analytic vector v of a unitary representation of G. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.

For the matrix coefficient π v,v (g)=π(g)v,v of a vector v in a unitary representation of an analytic Fréchet–Lie group G we show that v is an analytic vector if and only if π v,v is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a simply connected Fréchet–BCH–Lie group G extends to a global analytic function.

Soit G un groupe de Lie–Banach connexe et simplement connexe. Sur l’algèbre enveloppante complexe de son algèbre de Lie 𝔤 nous définissons la notion de fonctionnelle analytique et montrons que chaque fonctionnelle analytique positive λ est integrable au sens où elle est de la forme λ(D)=dπ(D)v,v pour un vecteur analytique v d’une représentation unitaire de G. Dans la preuve de ce résultat nous obtenons des critères pour l’integrabilité des *-representations des algèbres de Lie en représentations de groupe unitaires.

Pour le coefficient matriciel π v,v (g)=π(g)v,v d’un vecteur v d’une représentation unitaire d’un groupe de Lie–Fréchet analytique G nous montrons que v est un vecteur analytique si et seulement si π v,v est analytique dans un voisinage de l’identité. En combinant ce résultat à ceux sur les fonctionnelles analytiques positives nous obtenons que chaque fonction analytique de type positive locale sur un group de Lie–Fréchet–BCH simplement connexe s’étend en une fonction analytique globale.

DOI : https://doi.org/10.5802/aif.2660
Classification:  22E65,  22E45
Keywords: Infinite dimensional Lie group, unitary representation, positive definite function, analytic vector, integrability of Lie algebra representations.
@article{AIF_2011__61_5_1839_0,
     author = {Neeb, Karl-H.},
     title = {On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {1839-1874},
     doi = {10.5802/aif.2660},
     mrnumber = {2961842},
     zbl = {1241.22023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_1839_0}
}
Neeb, Karl-H. On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1839-1874. doi : 10.5802/aif.2660. http://www.numdam.org/item/AIF_2011__61_5_1839_0/

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