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.

The classical matter fields are sections of a vector bundle E with base
manifold M. The space L^2(E) of square integrable matter fields w.r.t. a
locally Lebesgue measure on M, has an important module action of C_b^\infty(M)
on it. This module action defines restriction maps and encodes the local
structure of the classical fields. For the quantum context, we show that this
module action defines an automorphism group on the algebra A, of the canonical
anticommutation relations on L^2(E), with which we can perform the analogous
localization. That is, the net structure of the CAR, A, w.r.t.

Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple
complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie
algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic
subgroup $P(\cA)$ of $G(\cA)$ and we obtain generalizations $X(\cA) :=
G(\cA)/P(\cA)$ of the generalized flag manifolds. In this note we provide an
explicit description of all homogeneous holomorphic line bundles over $X(\cA)$
with non-zero holomorphic sections.