Ebook: 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).

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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. Schm?dgen [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).

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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. Schm?dgen [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).
Content:
Front Matter....Pages i-xx
Front Matter....Pages 1-1
Unbounded Linear Operators in Hilbert Spaces....Pages 3-42
Partial O*-Algebras....Pages 43-95
Commutative Partial O*-Algebras....Pages 97-127
Topologies on Partial O*-Algebras....Pages 129-163
Tomita—Takesaki Theory in Partial O*-Algebras....Pages 165-253
Front Matter....Pages 255-255
Partial *-Algebras....Pages 257-300
*-Representations of Partial *-Algebras....Pages 301-337
Well-behaved *-Representations....Pages 339-365
Biweights on Partial *-Algebras....Pages 367-427
Quasi *-Algebras of Operators in Rigged Hilbert Spaces....Pages 429-468
Physical Applications....Pages 469-492
Outcome....Pages 493-494
Back Matter....Pages 495-521

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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. Schm?dgen [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).
Content:
Front Matter....Pages i-xx
Front Matter....Pages 1-1
Unbounded Linear Operators in Hilbert Spaces....Pages 3-42
Partial O*-Algebras....Pages 43-95
Commutative Partial O*-Algebras....Pages 97-127
Topologies on Partial O*-Algebras....Pages 129-163
Tomita—Takesaki Theory in Partial O*-Algebras....Pages 165-253
Front Matter....Pages 255-255
Partial *-Algebras....Pages 257-300
*-Representations of Partial *-Algebras....Pages 301-337
Well-behaved *-Representations....Pages 339-365
Biweights on Partial *-Algebras....Pages 367-427
Quasi *-Algebras of Operators in Rigged Hilbert Spaces....Pages 429-468
Physical Applications....Pages 469-492
Outcome....Pages 493-494
Back Matter....Pages 495-521
....