Tomita-Takesaki theory in algebras of unbounded operators (Q1273588)

The reviewed monograph is an extensive and systematic summary of the Tomita-Takesaki theory for von Neumann algebras generalized to the case of algebras \({\mathcal O}^*\) of unbounded operators in a Hilbert space. The theory has numerous applications in mathematics and theoretical physics, in particular, in the study of the structure of von Neumann algebras, and in the Wightman approach to quantum field theory. The note is divided into four chapters, devoted respectively to general theory of \({\mathcal O}^*\)-algebras, to standard generalized vectors and related standard and modular systems, to standard weights and related topics, and finally, to physical applications of the presented theory -- in particular to quantum statistical mechanics and the Wightman quantum field theory. An extensive bibliography completes this study. Concerning the completeness, systematic approach and many references to applications, the notes can be recommended to all interested in the theory and practice of systems of unbounded operators.