Partial \(^*\)-algebras and their operator realizations.

*(English)*Zbl 1023.46004
Mathematics and its Applications (Dordrecht). 553. Dordrecht: Kluwer Academic Publishers. xx, 521 p. (2002).

This monograph is the first comprehensive book on partial *-algebras and partial \(O^*\)-algebras. The book touches all aspects of this field and it presents the results in a very clearly arranged way. The majority of the results in this field originated from the authors and their students. Each chapter is complemented by notes including some details on the historical development.

The main motivation for the study of partial *-algebras of operators, called partial \(O^*\)-algebras for short, was the generalization of essential parts of the theory of \(C^*\)-algebras to algebras of unbounded operators on a Hilbert space. In this case, a product is always defined, and this leads to the notion of a partial multiplication, which reads axiomatically as follows:

Definition: \(A\) partial *-algebra is a complex vector space \({\mathcal A}\) with an involution \(x^*\to X\) satisfying \((x+\lambda y)^*=x^*+ \overline\lambda y^*\), and a partial multiplication \(\cdot:\Gamma \to{\mathcal A}\) defined on a domain \(\Gamma\subseteq {\mathcal A}\) satisfying the following axioms: (i) \((x,y)\in\Gamma\) iff \((y^*,x^*)\in\Gamma\), (ii) \((x,\lambda y+z)\in\Gamma\) for \((x,y)\in\Gamma\) and \((x,z)\in\Gamma\), (iii) \(x\cdot(y+ \lambda z)=x\cdot y+\lambda (x\cdot z)\) and \((x\cdot y)^*y^*\cdot x^*\), provided that \((x,y) \in\Gamma\) and \((x,z)\in\Gamma\).

As indicated in the title, the book is divided into two chapters: The theory of partial \(O^*\)-algebras in chapter one, and the *-representation of abstract partial *-algebras as partial \(O^*\)-algebras in chapter two. In the first part, after presenting several important examples, the following topics are covered: The extension of partial \(O^*\)-algebras to larger domains, the right definition of commutants and bicommutants, locally convex topologies on partial \(O^*\)-algebras, and finally the generalization of the Tomita-Takesaki theory. In the second part, the Gelfand-Naimark-Segal construction as a key of each representation theory is generalized to partial *-algebras. The book is completed by some physical applications. The headings of the sections are:

Chapter 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 \(O^*\)-algebras, 5. Tomita-Takesaki theory in partial \(O^*\)-algebras. Chapter II (Theory of partial *-algebras): 6. Partial *-algebras, 7. *-Representations of partial *-algebras, 8. Well-behaved *-representations, 9. Biweights on partial *-algebras, 10. Quasi *-algebras of operators in rigged Hilbert spaces, 11. Physical applications.

This very well written book is not only recommended to anybody interested in partial *-algebras, but also to those who want to deepen their knowledge of the classical theory of *-algebras and their representations. The bibliography is very extensive, comprising 21 pages.

The main motivation for the study of partial *-algebras of operators, called partial \(O^*\)-algebras for short, was the generalization of essential parts of the theory of \(C^*\)-algebras to algebras of unbounded operators on a Hilbert space. In this case, a product is always defined, and this leads to the notion of a partial multiplication, which reads axiomatically as follows:

Definition: \(A\) partial *-algebra is a complex vector space \({\mathcal A}\) with an involution \(x^*\to X\) satisfying \((x+\lambda y)^*=x^*+ \overline\lambda y^*\), and a partial multiplication \(\cdot:\Gamma \to{\mathcal A}\) defined on a domain \(\Gamma\subseteq {\mathcal A}\) satisfying the following axioms: (i) \((x,y)\in\Gamma\) iff \((y^*,x^*)\in\Gamma\), (ii) \((x,\lambda y+z)\in\Gamma\) for \((x,y)\in\Gamma\) and \((x,z)\in\Gamma\), (iii) \(x\cdot(y+ \lambda z)=x\cdot y+\lambda (x\cdot z)\) and \((x\cdot y)^*y^*\cdot x^*\), provided that \((x,y) \in\Gamma\) and \((x,z)\in\Gamma\).

As indicated in the title, the book is divided into two chapters: The theory of partial \(O^*\)-algebras in chapter one, and the *-representation of abstract partial *-algebras as partial \(O^*\)-algebras in chapter two. In the first part, after presenting several important examples, the following topics are covered: The extension of partial \(O^*\)-algebras to larger domains, the right definition of commutants and bicommutants, locally convex topologies on partial \(O^*\)-algebras, and finally the generalization of the Tomita-Takesaki theory. In the second part, the Gelfand-Naimark-Segal construction as a key of each representation theory is generalized to partial *-algebras. The book is completed by some physical applications. The headings of the sections are:

Chapter 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 \(O^*\)-algebras, 5. Tomita-Takesaki theory in partial \(O^*\)-algebras. Chapter II (Theory of partial *-algebras): 6. Partial *-algebras, 7. *-Representations of partial *-algebras, 8. Well-behaved *-representations, 9. Biweights on partial *-algebras, 10. Quasi *-algebras of operators in rigged Hilbert spaces, 11. Physical applications.

This very well written book is not only recommended to anybody interested in partial *-algebras, but also to those who want to deepen their knowledge of the classical theory of *-algebras and their representations. The bibliography is very extensive, comprising 21 pages.

Reviewer: H.Junek (Potsdam)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46L55 | Noncommutative dynamical systems |

47L60 | Algebras of unbounded operators; partial algebras of operators |

46L05 | General theory of \(C^*\)-algebras |

47L90 | Applications of operator algebras to the sciences |

46K05 | General theory of topological algebras with involution |