Synopses & Reviews
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).
Synopsis
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 Japanese mathematicians. This is the first textbook to go one step further by considering systematically partial *-algebras of unbounded operators (partial O*-algebras) and the underlying algebraic structure, namely, partial *-algebras. The first part of the text begins with partial O*-algebras covering basic properties and topologies with many examples and accumulates in 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 text then focuses on abstract partial *-algebras and their representation theory, again obtaining generalizations of familiar theorems, for example Radon-Nikodym and Lebesgue. Partial *-algebras of operators on Rigged Hilbert Spaces are also considered. The last chapter discusses some applications in mathematical physics, for example quantum field theory and spin systems. This book will be of interest to graduate students or researchers in pure mathematics as well as mathematical physicists.
Synopsis
This textbook considers systematically partial *-algebras of unbounded operators (partial O*-algebras) and the underlying algebraic structure, namely, partial *-algebras.
Table of Contents
Foreword. Introduction.
I: Theory of Partial O^{*}-Algebras. 1. Unbounded Linear Operators in Hilbert Spaces.
2. Partial O
^{*}-Algebras.
3.Commutative Partial O
^{*}-Algebras.
4. Topologies on Partial O
^{*}-Algebras.
5. Tomita Takesaki Theory in Partial O
^{*}-Algebras.
II: Theory of Partial ^{*}-Algebras. 6. Partial
^{*}-Algebras.
7. ^{*}-Representations of Partial
^{*}-Algebras.
8. Well-behaved X>*-Representations.
9. Biweights on Partial
^{*}-Algebras.
10. Quasi
^{*}-Algebras of Operators in Rigged Hilbert Spaces.
11. Physical Applications. Outcome. Bibliography. Index.