\end{split}$$

Theorem \[weaklocality2\] does not hold in the covering space of , since, in general, $e^{i\pi M_{0n}}$ and $J_{W_R}$ will not commute in such theories. Indeed, in [@F2; @BEM] can be found examples of a free field theory on the covering space of for which it can be shown that assumptions (i)–(iv) are satisfied, but the elements of $\As(W_R)$ and $\As(\widetilde{W}_R{}')$ do not commute in the vacuum state — see Appendix \[examples\] for further discussion. However, for theories on proper AdS, Theorem \[weaklocality2\] is consistent with a property of two-point functions observed in [@BEM] and arrived at there by very different means and assumptions.

In Theorems \[weaklocality1\] and \[weaklocality2\] we see that the passivity of the vacuum state and the group relations in ${\mbox{SO$_0(2,n\!-\!1)$}}$ have determined which regions in are to have (weakly) commensurable observables. In theories on AdS where the basic fields satisfy standard c–number commutation relations, it follows from this result that such observables actually commute in the usual (operator) sense [@StWi]. Indeed, it follows then that $$\label{locality}
\As(W_R ') \subset \As(W_R)' \quad \textnormal{and} \quad
\As(W_R) \subset \As(\widetilde{W}_R{}')' \, .$$ But assumptions (i)–(iv) do not imply the strong locality relations (\[locality\]) in general. Indeed, consider a tempered hermitian 