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Tomita conjugations and transitivity of locality

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Tomita conjugations and transitivity of locality ANNALES DE L’I. H. P., SECTION A JAKOB YNGVASON Tomita conjugations and transitivity of locality Annales de l’I. H. P., section A, tome 64, no 4 (1996), p. 395-408. <http://www.numdam.org/item?id=AIHPA_1996__64_4_395_0> © Gauthier-Villars, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A », implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/legal.php). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 395 Tomita conjugations and transitivity of locality Jakob YNGVASON Science Institute, University of Iceland Ann. Inst. Henri Poincare, Vol. 64, n° 4, 1996, Physique theorique ABSTRACT. - The Tomita conjugation, 5’, associated with a von Neumann algebra, and a cyclic and separating vector, H, is the closure of the map XS~ This notion can be generalized to algebras (and even more general families) of unbounded operators that appear in quantum field theory. It is shown that generalizations of the classical results of Borchers on transitivity of locality and of Bisognano and Wichmann on duality in quantum field theory follow from properties of such Tomita conjugations. The basis is the simple observation that two (unbounded) operator algebras with the same Tomita conjugation have the same unbounded weak commutant, provided one algebra is contained in the other. La conjugaison de Tomita, S’, associee a une algebre de von Neumann, .11~1, et a un vecteur cyclique et separateur, 0, est la fermeture de 1’ application X S2 ---~ X*Q, X E ./1~l . On peut generaliser cette notion aux algebres (et meme a des familles plus generales) d’ operateurs non bornés qui apparaissent dans la theorie quantique d

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