On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Abstract

Let G be a connected and simply connected Banach–Lie group or, more generally, a BCH–Lie group. On the complex enveloping algebra UC(g) 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 πv,v(g) = 〈π(g)v, v〉 of a vector v in a unitary representation of an analytic Frechet–Lie group G we show that v is an analytic vector if and only if πv,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 1-connected Frechet–BCH–Lie group G extends to a global analytic function.