  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.

Representation Theory;Operator Algebras