An unbounded generalization of the Tomita-Takesaki theory. II
We have continued our study of an unbounded generalization of the Tomita- Takesaki theory begun in a previous paper [ibid. 22, 725-765 (1986; Zbl 0624.47044)]. We first define the notion of modular vectors for a closed \(0\) \(*_ p\)-algebra (\({\mathcal M},{\mathcal D})\) and showed that if \(\xi_ 0\) is a modular vector for (\({\mathcal M},{\mathcal D})\) then a one-parameter group \(\{\sigma_ t^{\xi_ 0}\}\) of *-automorphisms of an unbounded biscommutant (\({\mathcal M}| {\mathcal D}_{\xi_ 0})''_{wc}\) of the \(0\) \(*_ p\)-algebra \({\mathcal M}| {\mathcal D}_{\xi_ 0}\) on a dense subspace \({\mathcal D}_{\xi_ 0}\) of \({\mathcal D}\) is defined and the vector state \(\omega_{\xi_ 0}\) on (\({\mathcal M}| {\mathcal D}_{\xi_ 0})''_{wc}\) satisfies the KMS-condition with respect to \(\{\sigma_ t^{\xi_ 0}\}.\) We have applied the unitary Radon-Nikodym cocycle introduced by Connes to the unbounded case. We have studied Radon-Nikodym theorems and Lebesgue decomposition theorems for \(0\) \(*_ p\)-algebras.
*- automorphisms of an unbounded biscommutant
Radon-Nikodym theorems and Lebesgue decomposition theorems for \(0\) \(*_ p\)-algebras
scientific article; zbMATH DE number 4044551
scientific article