Let be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra we define the concept of an analytic functional and show that every positive analytic functional is integrable in the sense that it is of the form for an analytic vector of a unitary representation of . 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 of a vector in a unitary representation of an analytic Fréchet–Lie group we show that is an analytic vector if and only if 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 extends to a global analytic function.
Soit un groupe de Lie–Banach connexe et simplement connexe. Sur l’algèbre enveloppante complexe de son algèbre de Lie nous définissons la notion de fonctionnelle analytique et montrons que chaque fonctionnelle analytique positive est integrable au sens où elle est de la forme pour un vecteur analytique d’une représentation unitaire de . 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 d’un vecteur d’une représentation unitaire d’un groupe de Lie–Fréchet analytique nous montrons que est un vecteur analytique si et seulement si 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.
Keywords: Infinite dimensional Lie group, unitary representation, positive definite function, analytic vector, integrability of Lie algebra representations.
Mot clés : groupe de Lie de dimension infinie, représentation unitaire, fonction de type positif, vecteur analytique, integrabilité d’une représentation d’une algèbre de Lie
@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}, pages = {1839--1874}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {5}, year = {2011}, doi = {10.5802/aif.2660}, zbl = {1241.22023}, mrnumber = {2961842}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2660/} }
TY - JOUR AU - Neeb, Karl-H. TI - On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups JO - Annales de l'Institut Fourier PY - 2011 SP - 1839 EP - 1874 VL - 61 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2660/ DO - 10.5802/aif.2660 LA - en ID - AIF_2011__61_5_1839_0 ER -
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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://archive.numdam.org/articles/10.5802/aif.2660/
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