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Let   ${{\mathfrak K}}\to X$   be a smooth Lie algebra bundle over a   $\sigma$  -compact manifold   $X$   whose typical fiber is the compact Lie algebra   ${{\mathfrak k}}$  . We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Fréchet–Lie algebra   $\Gamma ({{\mathfrak K}})$   of all smooth sections of   ${{\mathfrak K}}$  , and of the LF-Lie algebra   $\Gamma _c({{\mathfrak K}})$   of compactly supported smooth sections. For   $\Gamma ({{\mathfrak K}})$  , irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For   $\Gamma _{c}({{\mathfrak K}})$  , 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   ${{\mathfrak k}}\otimes _{\mathbb {R}} {\mathcal {A}}_{\mathbb {R}}$  , where   ${\mathcal {A}}_{\mathbb {R}}$   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   ${{{\rm Gau}}}(P)_0$   of the group of smooth gauge transformations of a principal fiber bundle   $P \to X$   with compact base and structure group, and the groups   ${{{\rm SU}}}_n({\mathcal {A}})_0$   with   ${\mathcal {A}}$   a complete involutive commutative continuous inverse algebra.

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections