1. Partial *- Algebras and Their Operator Realizations
- Author
-
J-P Antoine, I. Inoue, C. Trapani, J-P Antoine, I. Inoue, and C. Trapani
- Subjects
- Mathematical analysis, Operator theory, Mathematics
- Abstract
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).
- Published
- 2013