Tomita—Takesaki Theory in Partial O*-Algebras

Abstract

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 EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacq % aH7oaBaaaaaa!37B9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\overline \lambda$$ for the partial GW*-algebra (M ^’_w ) ^’ _σ ., in such a way that λ^c = EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaacq % aH7oaBaaWaaWbaaSqabeaacaWGJbaaaaaa!38CE!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\overline \lambda ^c}$$. 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 ^λ _t }t∈ℝ of the partial O*-algebra M; then the generalized vector λ satisfies the KMS condition with respect to {σ ^λ _t }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);