Partial *- Algebras and Their Operator Realizations

Algebras of bounded operators are familiar, either as C ^* -algebras or as von Neumann algebras. A first generalization is the notion of algebras of unbounded operators (O ^* -algebras), mostly developed by the Leipzig school and in Japan (for a review, we refer to the monographs of K. Schmüdgen [1990] and A. Inoue [1998]). This volume goes one step further, by considering systematically partial ^* -algebras of unbounded operators (partial O ^* -algebras) and the underlying algebraic structure, namely, partial ^* -algebras. It is the first textbook on this topic.
The first part is devoted to partial O ^* -algebras, basic properties, examples, topologies on them. The climax is the generalization to this new framework of the celebrated modular theory of Tomita-Takesaki, one of the cornerstones for the applications to statistical physics.
The second part focuses on abstract partial ^* -algebras and their representation theory, obtaining again generalizations of familiar theorems (Radon-Nikodym, Lebesgue).