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Tomita—Takesaki Theory in Partial O*-Algebras

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This chapter is devoted to the development of the Tomita-Takesaki theory in partial O*-algebras. In Section 5.1, we introduce and investigate the notion of cyclic generalized vectors for a partial O*-algebra, generalizing that of cyclic vectors, and its commutants. Section 5.2 introduces the notion of a cyclic and separating system (M, λ, λ c ), which consists of a partial O*-algebra M, a cyclic generalized vector λ for M and the commutant λ c of λ. A cyclic and separating system (M, λ, λ c ) determines the cyclic and separating system ((M w ′ )′, λ cc , (λ cc ) c ) of the von Neumann algebra (M w ′ )′, and this makes it possible to develop the Tornita-Takesaki theory. Then λ can be extended to a cyclic and separating generalized vector for the partial GW*-algebra (M w ’ ) σ ’ ., in such a way that λ c = . Section 5.3 develops the Tornita fundamental theorem according to the method of Van Daele. Section 5.4 introduces the notion of standard generalized vectors for a partial O*-algebra, which enables one to develop the Tomita-Takesaki theory in partial O*-algebras. Given a standard generalized vector λ for a partial O*-algebra M, one constructs the one-parameter group of *-automorphisms {σ t λ }t∈ℝ of the partial O*-algebra M; then the generalized vector λ satisfies the KMS condition with respect to {σ t λ }t∈ℝ . Section 5.5 introduces the notion of modular generalized vectors for a partial O*-algebra, which gives rise to standard generalized vectors for a partial GW*-algebra. Section 5.6 deals with some particular cases of standard or modular generalized vectors for partial O*-algebras (generalized vectors associated to individual vectors (Section 5.6.1);