By Halvorson, Mueger.

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Then there is a unique type I factor N such that: (i) R1 ⊆ N ⊆ R2 , and (ii) N is invariant under all automorphisms of R2 that preserve both R1 and the state ω. Of course, the algebra N itself does not have dispersion-free states, and so cannot be the algebra of definite observables. However, the state ω|N is normal, and since N is a type I factor, there is a density operator D ∈ N that induces the state in the sense that ω(A) = Tr(DA), for all A ∈ N. Then assuming that R1 must for some reason be left invariant under symmetries of R2 , the algebra D = {D} looks like a good candidate for the modal interpreter’s set of definite-value observables in R2 in the state ω.

Xn ∈ Rn , we have n n cj ci f (xi − xj ) ≥ 0. 3 Theorem (Bochner). Let f : Rn → C be a continuous function of positive type. Then the Fourier transform of f is a bounded measure on Rn . 44 Proof. For a proof of Bochner’s theorem, see [Rudin, 1991, p. 303] and [Folland, 1995, p. 95]. 4 Remark. Only the group structure of Rn is really needed for Bochner’s theorem. So, we are not making any mistake by thinking of Minkowski spacetime as R4 in this context. We will need the following key lemma in all of our subsequent results.

See [Halvorson and Clifton, 2000]. 22 Remark. If a net O → R(O) of von Neumann algebras on H satisfies property B and nontriviality, then the hypotheses of Prop. 21 apply to algebras R(O1 ) and R(O2 ) when O1 and O2 are strongly spacelike separated. Notes: For a comprehensive review of pre-1990 results on independence of local algebras in AQFT, see [Summers, 1990]. For some more recent results, see [Summers, 1997; Florig and Summers, 1997; R´edei, 1998; Halvorson and Clifton, 2000; Summers and Buchholz, 2005].