$\lambda \in {\mbox{SO$_0(2,n\!-\!1)$}}$. Hence, since $\As(W_R{}') = e^{i\pi M_{12}}\As(W_R)e^{-i\pi M_{12}}$, equation (\[unique\]) and the observations made above imply $$J_{W_R{}'} = e^{i\pi M_{12}} J_{W_R} e^{-i\pi M_{12}} = 
J_{W_R} e^{-i2\pi M_{12}} = J_{W_R} \, .$$ Thus, the ${\mbox{SO$_0(2,n\!-\!1)$}}$-covariance of the net entails $$\label{equation15}
J_{W'} = J_{W} \, , \quad \textnormal{for every $W \in \Ws$} \, .$$ If the algebras $\As(W)$ and $\As(W')$ commuted strongly with each other, it can be shown that (\[equation15\]) follows from modular theory. It is therefore of interest that (\[equation15\]) obtains when the algebras only weakly commute.

We continue with some further locality results, which distinguish theories on proper AdS from those on the covering space. We have chosen our wedge regions $W \in \Ws$ to be connected for physical reasons, as previously explained. But the intersection of the (connected) wedge [**W**]{} in the ambient space $${\bf W} = \{ x \in \RR^{n+1} \mid x_1 > |x_0| \}$$ with has two connected components, one of which is $W_R$ and the other is the conjugate wedge $$\widetilde{W}_R{}' = \{ x \in {\mbox{{AdS}$^n\,$}}\mid x_1 > |x_0|\, , -x_n > 0 \} \, .$$ One has the geometric relations

\[F1\]

$$\widetilde{W}_R = \theta_{01}\widetilde{W}_R{}^\prime = -W_R \quad 
\textrm{and} \quad W_R{}' = -\widetilde{W}_R{}' 