Integrating Unitary Representations of Infinite-Dimensional Lie Groups

Abstract

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.