Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Bas Janssens*, Karl-Hermann Neeb

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Let K→X be a smooth Lie algebra bundle over a σ-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 Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K)
of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional.
For Γ_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⊗A, where A 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 SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.
Original languageEnglish
Pages (from-to)9081-9137
JournalInternational Mathematics Research Notices
Volume2015
Issue number18
Early online date1 Dec 2014
DOIs
Publication statusPublished - 2015

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