PCT–theorem can be proven.

\[proper\] If Standing Assumptions (i)–(iv) hold, then the unitary representation $U$ of extends to a representation of in which the reflection $\theta_{01}$ is implemented by the anti–unitary involution $J_{W_R}$.

Note that ${\mbox{SO$(2,n\!-\!1)$}}$ is the disjoint union of $\theta_{01}{\mbox{SO$_0(2,n\!-\!1)$}}$ and ${\mbox{SO$_0(2,n\!-\!1)$}}$. Since $$U(\theta_{01} \lambda_1 \cdot \lambda_2) = J_{W_R} U(\lambda_1 \lambda_2)
= J_{W_R} U(\lambda_1)U(\lambda_2) = U(\theta_{01}\lambda_1)U(\lambda_2)$$ and $$\begin{aligned}
U(\theta_{01} \lambda_1 \cdot \theta_{01}\lambda_2) & = &
U(\theta_{01} \lambda_1 \theta_{01})U(\lambda_2) \\ \nonumber
& = & J_{W_R} U(\lambda_1) J_{W_R} U(\lambda_2)
= U(\theta_{01}\lambda_1)U(\theta_{01}\lambda_2) \, ,\end{aligned}$$ for all $\lambda_1,\lambda_2 \in {\mbox{SO$_0(2,n\!-\!1)$}}$, the assertion follows.

Theorem \[proper\] is a purely group–theoretic statement which does not yet say anything about the adjoint action of $J_{W_R}$ on the observables. Results of that type require an additional assumption and will appear in a later publication.

Weak locality {#weaklocalitysec}
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In order to gain insight into the locality properties of the net, we consider the observables which are localized in the region $$W_R{}^{\prime} \doteq \{x\in{\mbox{{AdS}$^n\,$}}\mid - x_1 > | x_0 |, \, x_n > 0\} \, .$$ Throughout this subsection we shall only consider of dimension $n \geq 3$. Since the regions $W_R$ and $W_R{}^{\prime}$ are each one–half (on the same “side” of ${\mbox{{AdS}$^n\,$}}$ — see Figure 1 