\As(\widetilde{W}_1{}')$.

The Schlieder property {#Schliedersec}
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Many versions of the notion of independence of algebras of observables in spacelike separated regions have emerged in algebraic quantum theory (see [@Sum] for a review), and most are logically independent of the usual notion of commensurability, which is that the algebras commute with each other elementwise. In this section we shall prove that algebras associated with properly spacelike separated wedges $W_1,W_2$ satisfy an extended form of the algebraic independence condition known as the Schlieder property, namely that $A_1 \in \As(W_1), A_2 \in \As(W_2)$ and $A_1 A_2 = 0$ imply either $A_1 = 0$ or $A_2 = 0$. We shall say that two wedges $W_1,W_2$ are properly spacelike separated if $\lambda W_1 \subset W_2{}'$ for all $\lambda$ in some neighborhood of the identity in . Note that $W$ and $W'$ are not properly spacelike separated. Although in de Sitter and Minkowski spaces of dimension $n \geq 3$ such properly spacelike separated wedges do not exist, we show in Appendix \[sectioninclusion\] that they are plentiful in AdS.

The proof of an extended Schlieder property in , $n \geq 3$, will be carried out in a series of steps.

\[indprepare\] Let $W_1,W_2 \in \Ws$ be properly 