  We show that in the presence of suitable commutator estimates, a projective
unitary representation of the Lie algebra of a connected and simply connected
Lie group G exponentiates to G. Our proof does not assume G to be
finite--dimensional or of Banach-Lie type and therefore encompasses the
diffeomorphism groups of compact manifolds. We obtain as corollaries short
proofs of Goodman and Wallach's results on the integration of positive energy
representations of loop groups and Diff(S^{1}) and of Nelson's criterion for
the exponentiation of unitary representations of finite-dimensional Lie
algebras.
