Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I

Abstract

Let $(\pi ,\mathcal{H})$ be a strongly continuous unitary representation of a 1-connected Lie group $G$ such that the Lie algebra $\mathfrak{g}$ of $G$ is generated by the positive cone $C_{\pi }:=\{x\in \mathfrak{g}:-i\partial \pi (x)\ge 0\}$ and an element $h$ for which the adjoint representation of $h$ induces a 3-grading of $\mathfrak{g}$. Moreover, suppose that $(\pi ,\mathcal{H})$ extends to an antiunitary representation of the extended Lie group $G_{\tau }:=G⋊\{1,{\tau }_G\}$, where ${\tau }_G$ is an involutive automorphism of $G$ with $\mathbf{L}({\tau }_G)=e^{i\pi \mathrm{ad}h}$. In a recent work by Neeb and Ólafsson, a method for constructing nets of standard subspaces of $\mathcal{H}$ indexed by open regions of $G$ has been introduced and applied in the case where $G$ is semisimple. In this paper, we extend this construction to general Lie groups $G$, provided the above assumptions are satisfied and the center of the ideal ${\mathfrak{g}}_C=C_{\pi }-C_{\pi }$ of $\mathfrak{g}$ is one-dimensional. The case where the center of ${\mathfrak{g}}_C$ has more than one dimension will be discussed in a separate paper.

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