is a factor (cf. the proof of Theorem \[split\]). If wedge locality (\[locality\]) held, then it would follow that $\As(W) \cap \As(W') = \CC 1$, [*i.e.*]{} no nontrivial observable can be localized in both $W$ and $W'$. It is therefore of physical interest that this fact also follows directly from (i)–(iv) in the absence of (\[locality\]).

\[extendedlocality\] Let Standing Assumptions (i)–(iv) hold and let $n \geq 3$. Then for any $W \in \Ws$, one has $\As(W) \cap \As(W') = \CC 1$ and $\As(W) \bigvee \As(W') = \Bs(\Hs)$.

By covariance, it suffices to prove the assertion for $W = W_R$. Let $A \in \As(W_R) \cap \As(W_R{}')$ and $\Hs_0$ be the closure in $\Hs$ of $\big( \As(W_R) \cap \As(W_R{}') \big)\Omega$. Since $e^{i2t\pi M_{01}}$, respectively $e^{-i2t\pi M_{01}}$, are the modular unitaries corresponding to $(\As(W_R),\Omega)$, respectively $(\As(W_R{}'),\Omega)$, then with $\Delta = e^{2\pi M_{01}}$ one has $$J_{W_R} \Delta^{1/2} A \Omega = A^* \Omega \, , \quad  
J_{W_R{}'} \Delta^{-1/2} A \Omega = A^* \Omega \, .$$ Hence, equation (\[Js\]) entails that $\Delta^{1/2} = \Delta^{-1/2}$ on $\Hs_0$. Therefore, $\Delta A \Omega = A \Omega$, which yields $A \Omega = \Delta^{it} A \Omega = e^{i2t\pi M_{01}} A \Omega$, for all $t \in \RR$. Hence, the Mean Ergodic Theorem 