Mathematics

The theory of M. Tomita of the standard form of general von Neumann algebras was a turning point in the theory of Operator Algebras and is up to this day one of the most important tools by working with von Neumann algebras. It became accessible in 1970 in the exposition of M. Takesaki, which contains so many fundamental contributions also of him that the whole theory is usually referred as the "Tomita-Takesaki Theory".The Tomita-Takesaki Theory is quite involved and can be contemplated from different points of view. In the decade 1970 - 1980 several approaches appeared to the theory, each one seeking to attain more transparency. One of them was the paper of S. L. Woronowicz "Operator Systems and their application to the Tomita-Takesaki Theory" appeared in 1979.In this talk we shall present a personal version of Woronowicz's approach to the Tomita-Takesaki Theory. The fundamental facts of the theory will be deduced by applying an original implementation theorem for one-parameter *-automorphism groups of von Neumann algebras, due to Woronowicz. Its applicability is made possible by S. Sakai's polar decomposition theorem for linear forms, as well as by half of the proof of his Radon-Nikodym type theorem.