Norm continuous unitary representations of Lie algebras of smooth sections
Norm continuous unitary representations of Lie algebras of smooth sections
复制标题
光滑截面李代数的范数连续酉表示
DOI：
10.1093/imrn/rnu231
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
K. Neeb
中科院分区：
文献类型：
--
作者：
B. Janssens;K. Neeb
给出了典型纤维为紧李代数的光滑李代数丛的所有光滑截面的Fr‘echet-Lie代数以及紧支撑光滑截面的LF-Lie代数的有界(即范数连续)酉表示的完整刻划.对于所有截面的李代数，有界酉不可约表示是所谓的求值表示的有限张量积，因此特别是有限维。对于紧支撑截面的李代数，有界酉不可约(因子)表示可能是赋值表示的无限张量积，从而将分类问题归结为关于UHF C*-代数的不可约(因子)表示的Glimm和幂的结果。我们证明的关键部分是李代数的不可约有界酉性表示的分类，这些表示是紧李代数和单位实连续逆代数的张量积：每个这样的表示都是赋值表示的有限积。在群水平上，我们的结果特别地涵盖了具有紧基和结构群的主纤维丛的光滑规范变换群的单位分量的有界么正表示，以及值在对合连续逆代数中的特殊酉阵的群的连通分量的有界酉性表示。
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.