Janssens B, Neeb KH (2015)
Publication Type: Journal article, Original article
Publication year: 2015
Publisher: Oxford University Press (OUP): Policy H - Oxford Open Option A
Pages Range: 9081 - 9137
Journal Issue: 18
DOI: 10.1093/imrn/rnu231
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 Fréchet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γc(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 Γc(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⊗RAR, where AR 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.
APA:
Janssens, B., & Neeb, K.H. (2015). Norm continuous unitary representations of Lie algebras of smooth sections. International Mathematics Research Notices, 18, 9081 - 9137. https://dx.doi.org/10.1093/imrn/rnu231
MLA:
Janssens, Bas, and Karl Hermann Neeb. "Norm continuous unitary representations of Lie algebras of smooth sections." International Mathematics Research Notices 18 (2015): 9081 - 9137.
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