  Let g be a Banach Lie algebra and \tau : g ---> g an involution. Write g=h+q
for the eigenspace decomposition of g with respect to \tau and g^c := h+iq for
the dual Lie algebra. In this article we show the integrability of two types of
infinitesimally unitary representations of g^c. The first class of
representation is determined by a smooth positive definite kernel K on a
locally convex manifold M. The kernel is assumed to satisfying a natural
invariance condition with respect to an infinitesimal action \beta : g \to V(M)
by locally integrable vector fields that is compatible with a smooth action of
a connected Lie group $H$ with Lie algebra h.
  The second class is constructed from a positive definite kernel corresponding
to a positive definite distribution K \in C^{-\infty}(M \times M) on a finite
dimensional smooth manifold M which satisfies a similar invariance condition
with respect to a homomorphism \beta : g \to V(M). As a consequence, we get a
generalization of the Luscher--Mack Theorem which applies to a class of
semigroups that need not have a polar decomposition. Our integrability results
also apply naturally to local representations and representations arising in
the context of reflection positivity.
