POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS
POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS
复制标题
一般拓扑群的正能量表示和射影表示的连续性
作者：
K. Neeb
摘要设G和T为拓扑群，α: T→Aut(G)是定义T对G的连续作用的同态，G♯：= G <s:1> αT是对应的半直积群。本文讨论了G的不可约连续酉表示（π♯，${\mathcal{H}}$）对G的限制仍然不可约的几个问题。首先，我们证明，对于T = ${\mathbb R}$，对于任何不可约的G♯的正能量表示，即对于单参数群Ut:= π♯（1，T）具有非负谱，这是成立的。从G的不可约酉表示到g#的表示的过渡要求某些射影酉表示是连续的。为了便于验证，我们导出了射影酉表示连续性的各种有效准则。基于W*-动力系统的Borchers的结果，我们也得到了G上扩展到g#的连续正定函数的一个刻画。
Abstract Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G♯ := G ⋊αT the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯, ${\mathcal{H}}$) of G♯ whose restriction to G remains irreducible. First, we prove that, for T = ${\mathbb R}$, this is the case for any irreducible positive energy representation of G♯, i.e. for which the one-parameter group Ut := π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G♯ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G♯.