On analytic vectors for unitary representations of infinite dimensional lie groups

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 λ is integrable in the sense that it is of the form λ(D) = {dφ(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 φ v,v(g) = {φ(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 φ 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. © Association des Annales de l'institut Fourier, 2011, tous droits réservés.