On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Karl-H. Neeb[1]

  • [1] Department Mathematik, FAU Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054-Erlangen, Germany

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 1839-1874
  • ISSN: 0373-0956

Abstract

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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 ) = 〈 d π ( 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.

How to cite

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Neeb, Karl-H.. "On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups." Annales de l’institut Fourier 61.5 (2011): 1839-1874. <http://eudml.org/doc/219711>.

@article{Neeb2011,
abstract = {Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $\mathfrak\{g\}$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda $ is integrable in the sense that it is of the form $\lambda (D) = \langle \{\tt d\}\pi (D)v, v\rangle $ 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 $\pi ^\{v,v\}(g) = \langle \pi (g)v,v\rangle $ 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 $\pi ^\{v,v\}$ 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.},
affiliation = {Department Mathematik, FAU Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054-Erlangen, Germany},
author = {Neeb, Karl-H.},
journal = {Annales de l’institut Fourier},
keywords = {Infinite dimensional Lie group; unitary representation; positive definite function; analytic vector; integrability of Lie algebra representations; Lie group; Lie algebra; integrable representation},
language = {eng},
number = {5},
pages = {1839-1874},
publisher = {Association des Annales de l’institut Fourier},
title = {On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups},
url = {http://eudml.org/doc/219711},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 1839
EP - 1874
AB - Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $\mathfrak{g}$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda $ is integrable in the sense that it is of the form $\lambda (D) = \langle {\tt d}\pi (D)v, v\rangle $ 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 $\pi ^{v,v}(g) = \langle \pi (g)v,v\rangle $ 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 $\pi ^{v,v}$ 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.
LA - eng
KW - Infinite dimensional Lie group; unitary representation; positive definite function; analytic vector; integrability of Lie algebra representations; Lie group; Lie algebra; integrable representation
UR - http://eudml.org/doc/219711
ER -

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