Norm continuous unitary representations of Lie algebras of smooth sections

TL;DR

This paper classifies all bounded unitary representations of Lie algebras of smooth sections, revealing that they are constructed from evaluation representations, with implications for gauge groups and matrix groups.

Contribution

It provides a complete classification of bounded unitary representations of Lie algebras of smooth sections, extending to gauge groups and matrix groups, and introduces new structural results for tensor product representations.

Findings

Bounded irreducible representations are finite tensor products of evaluation representations.

For compactly supported sections, representations can be infinite tensor products, linking to UHF C*-algebra representations.

Every irreducible representation of tensor products with a compact Lie algebra and a continuous inverse algebra is a finite product of evaluation representations.

Abstract

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fr\'echet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, 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 classification of irreducible bounded unitary representations of Lie algebras that are the tensor product of a compact Lie algebra and a unital real continuous inverse algebra: every such representation is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component of the group of smooth gauge transformations of a principal fiber bundle with compact base and structure group, and the connected component of the group of special unitary n times n matrices with values in an involutive commutative continuous inverse algebra.