Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K -> X be a smooth Lie algebra bundle over a sigma-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i. e., norm continuous) unitary representations of the Frechet-Lie algebra Gamma (K) of all smooth sections of K, and of the LF-Lie algebra Gamma(c)(K) of compactly supported smooth sections. For Gamma (K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Gamma(c)(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C*-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k circle times(R) A(R), where A(R) is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)(0) of the group of smooth gauge transformations of a principal fiber bundle P -> X with compact base and structure group, and the groups SUn(A)(0) with A a complete involutive commutative continuous inverse algebra.