do not assume that all observables can be constructed out of observables with arbitrarily small localization region. This is because there are nets of physical interest on curved space–times for which the condition (v) specified below holds but the algebras $\As(\Os)$ associated with all bounded open regions $\Os$ are trivial [@BMS; @Rehren]. Also, there exist examples in which the algebra $\As(\Os)$ is nontrivial only for sufficiently large bounded regions $\Os$ [@BMS]. In both of these cases the assumption made in [@AdS] is violated. We have therefore eliminated all assumptions referring to bounded regions.

We emphasize that we do not postulate from the outset any local commutation relations of the observables. For, in contrast to the case of globally hyperbolic space–times, the principle of Einstein causality does not provide any clues as to which observables in AdS should commute with each other. Instead, we shall [*derive*]{} such commutation relations from stability properties of the vacuum, which we now specify.

We shall assume that the state $\omega$ determined by $\Omega$ is passive (cf. [@PuWo] and Section 5.4.4 in [@BratRob]) for the dynamical system $(\As(W), \ad U(\lambda(t)))$, for all geodesic and all uniformly accelerated observers described above. We recall that passivity is an 