POSITIVE ENERGY REPRESENTATIONS AND CONTINUITY OF PROJECTIVE REPRESENTATIONS FOR GENERAL TOPOLOGICAL GROUPS

Abstract

AbstractLetGandTbe topological groups, α :T→ Aut(G) a homomorphism defining a continuous action ofTonGandG♯:=G⋊αTthe corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯,${\mathcal{H}}$) ofG♯whose restriction toGremains irreducible. First, we prove that, forT=${\mathbb R}$, this is the case for any irreducible positive energy representation ofG♯, i.e. for which the one-parameter groupUt:= π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations ofGto representations ofG♯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 forW*-dynamical systems, we also derive a characterization of the continuous positive definite functions onGthat extend toG♯.