On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\C(\g)$ 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 $\lambda(D) = \la \dd\pi(D)v, v\ra$ 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 $\pi^{v,v}(g) = \la \pi(g)v,v\ra$ of a vector $v$ in a unitary representation of an analytic Fr\'echet-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 1-connected Fr\'echet--BCH--Lie group $G$ extends to a global analytic function.