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On analytic vectors for unitary representations of infinite dimensional Lie groups

Neeb KH (2011)

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Publication Type: Journal article, Original article

Publication year: 2011

Journal

Publisher: Association des Annales de l'Institute Fourier; 1999

Book Volume: 61

Pages Range: 1441 - 1476

Journal Issue: 5

URI: http://eudml.org/doc/219711

Abstract

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 $\lambda$ is integrable in the sense that it is of the form $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle$ 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 ${\pi }^{v,v}\left(g\right)=\langle \pi \left(g\right)v,v\rangle$ 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.

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How to cite

APA:

Neeb, K.H. (2011). On analytic vectors for unitary representations of infinite dimensional Lie groups. Annales de l'Institut Fourier, 61(5), 1441 - 1476.

MLA:

Neeb, Karl Hermann. "On analytic vectors for unitary representations of infinite dimensional Lie groups." Annales de l'Institut Fourier 61.5 (2011): 1441 - 1476.

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