Nets of standard subspaces on Lie groups

Understand

Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $\tau_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{\pi i ad(h)}$.

• We consider antiunitary representations $U$ of the Lie group $G_\tau = G \rtimes \{e,\tau_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$.
• To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspace $H_E(O) \subseteq H$, generated by the subspaces $U(\varphi)E$, where $\varphi \in C^\infty_c(O,\mathbb{R})$ is a real-valued test function on $O$.
• Then $H_E(O)$ is dense in $H_E(G)$ for every non-empty open subset $O \subseteq G$ (Reeh--Schlider property).