Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I
英文摘要：
Let $(π, \mathcal{H})$ 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 $C_π:= \{x \in \mathfrak{g} : -i\partial π(x) \geq 0\}$ and an element $h$ for which the adjoint representation of $h$ induces a 3-grading of $\mathfrak{g}$. Moreover, suppose that $(π, \mathcal{H})$ extends to an antiunitary representation of the extended Lie group $G_τ:= G \rtimes \{\mathbf{1}, τ_G\}$, where $τ_G$ is an involutive automorphism of $G$ with $\mathbf{L}(τ_G) = e^{iπ\mathrm{ad} h}$. 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 $\mathfrak{g}_C = C_π- C_π$ 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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