Standard Generalized Vectors for Partial O*-algebras
نویسندگان
چکیده
Standard generalized vectors for an O*-algebra N lead to a Tomita-Takesaki theory of modular automorphisms on N, and this is a key step in constructing KMS states on N (which would represent equilibrium states if N is the observable algebra of some physical system). In this paper, we extend to partial O*-algebras the notion of standard generalized vector and we show that they indeed satisfy the KMS condition. We also discuss several less restrictive classes of generalized vectors for a partial O*-algebra M, which all give rise to standard generalized vectors for a partial GW*-algebra canonically associated to M, either on the same domain or on a smaller dense domain. Finally we discuss the extension of standard generalized vectors from a von Neumann algebra A to a suitable partial GW*-algebra containing A. Thus here also partial GW*-algebras play a distinguished role among all partial O*-algebras. R esum e Les vecteurs g en eralis es standard pour une O*-alg ebre N m enent a une th eorie des automorphismes modulaires sur N, au sens de Tomita-Takesaki, et ceci est une etape cruciale dans la construction d' etats KMS sur N (qui repr esentent les etats d' equilibre si N est l'alg ebre des observables d'un syst eme physique). Dans le pr esent travail, on etend aux O*-alg ebres partielles la notion de vecteurs g en eralis es standard et on montre qu'ils v eriient eeectivement la condition KMS. On discute egalement dii erentes classes moins restrictives de vecteurs g en eralis es pour une O*-alg ebre partielle M, qui toutes donnent lieu a des vecteurs g en eralis es standard pour une GW*-alg ebre partielle canoniquement associ ee a M, tant^ ot sur le m^ eme domaine, tant^ ot sur un domaine dense plus petit. Ennn, etant donn e une alg ebre de von Neumann A, on discute le probl eme de l'extension des vecteurs g en eralis es standard pour A a une GW*-alg ebre partielle convenable contenant A. Il appert donc qu'ici aussi les GW*-alg ebres partielles jouent un r^ ole privil egi e parmi toutes les O*-alg ebres partielles.
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