Why the elements of a local von Neumann algebra cannot be analytic elements of the timelike translation?

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    $\begingroup$ Could you please state the definition of analytic element with respect to timelike translations? $\endgroup$ – Valter Moretti Aug 22 at 17:23
  • $\begingroup$ The definition is as found in Brattelli Robinson I: $\endgroup$ – val 72 Aug 22 at 18:47
  • $\begingroup$ I cannot check it now, but I guess that $X$ is the local von Neumann algebra and the indicated topology is one of the natural weak topologies interesting for von Neumann algebras. I think that one is also assuming some spectral condition on the representation of the group of four-translations implying that the generator of time displacements is bounded below. I expect that this asymmetry is responsible for the lack of analiticy in any set of the form $I_\lambda$. $\endgroup$ – Valter Moretti Aug 22 at 19:22
  • $\begingroup$ Yes, you are right! The question is about whether one, given the usual axioms of a Haag - Araki theory (Isotony, weak additivity, Poincare Covariance, Microcausality, Spectrum Condition) can think of an analytic element as belonging in a local algebra associated with an open bounded region, say, a double cone. $\endgroup$ – val 72 Aug 22 at 19:26
  • $\begingroup$ In general, X is the global von Neumann Algebra, so as to have the unitary implementation of the translations. However, can we consider the observable to belong in a local algebra? $\endgroup$ – val 72 Aug 22 at 19:29

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