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**Unbounded operator algebras and representation theory.
Lizenzausg. d. Akademie Verl., Berlin.**
*(English)*
Zbl 0697.47048

Operator Theory: Advances and Applications, 37. Basel etc.: Birkhäuser Verlag. 380 p. DM 148.00 (1989).

Algebraic systems of unbounded operators occur naturally in unitary representation theory of Lie groups as well as in the Wightman formulation of quantum field theory, but also in the theory of locally convex spaces, of von Neumann algebras, of distributions, of noncommutative probability theory.

This book provides with a treatment of *-algebras of unbounded operators in Hilbert spaces (0\({}^*\)-algebras) and of unbounded *-representations of *-algebras. An \(0^*\)-algebra is a *-algebra of unbounded linear operators on a common dense linear subspace of a Hilbert space leaving this subspace invariant. A *-representation of a *-algebra is a *- homomorphism of the *-algebra onto some \(0^*\)-algebra.

First of all, this book gives a treatment of the basic concepts involved in the theory: graph topology, closed and selfadjoint *-representations, closed and selfadjoint \(0^*\)-algebras, weak and strong commutant, strongly positive and completely strongly positive *-representations.

Then the developments of the theory are explained and proved, with the help of many examples and counter examples.

This book deals with a rather recent and in full expansion theory and is written by one of the most active researchers in the field.

Apart from a preliminary chapter, the book consists of two rather independent parts. The first one is devoted to the study of \(0^*\)- algebras and related topologies. In chapter 2 are introduced basic notions on \(0^*\)-algebras and the graph topology on the domain. Chapter 3 and 4 study various topologies on \(0^*\)-algebras. Chapter 5 deals with linear functionals defined by trace-class operators. In Chapter 6 are studied two special *-algebras. Finally Chapter 7 is concerned with commutants of \(0^*\)-algebras and makes the transition with the second part.

This second part is concerned with *-representations of *-algebras by unbounded operators. After some general concept developed in Chapter 8, Chapter 9 and 10 specialize to integrable representations of commutative *-algebras and of enveloping algebras. In Chapter 11 n-positive and completely positive *-representations are considered while Chapter 12 is concerned with integral decomposition of *-representations and states.

Each chapter ends with notes and bibliographical comments, while an up to date bibliography closes the book.

This book provides with a treatment of *-algebras of unbounded operators in Hilbert spaces (0\({}^*\)-algebras) and of unbounded *-representations of *-algebras. An \(0^*\)-algebra is a *-algebra of unbounded linear operators on a common dense linear subspace of a Hilbert space leaving this subspace invariant. A *-representation of a *-algebra is a *- homomorphism of the *-algebra onto some \(0^*\)-algebra.

First of all, this book gives a treatment of the basic concepts involved in the theory: graph topology, closed and selfadjoint *-representations, closed and selfadjoint \(0^*\)-algebras, weak and strong commutant, strongly positive and completely strongly positive *-representations.

Then the developments of the theory are explained and proved, with the help of many examples and counter examples.

This book deals with a rather recent and in full expansion theory and is written by one of the most active researchers in the field.

Apart from a preliminary chapter, the book consists of two rather independent parts. The first one is devoted to the study of \(0^*\)- algebras and related topologies. In chapter 2 are introduced basic notions on \(0^*\)-algebras and the graph topology on the domain. Chapter 3 and 4 study various topologies on \(0^*\)-algebras. Chapter 5 deals with linear functionals defined by trace-class operators. In Chapter 6 are studied two special *-algebras. Finally Chapter 7 is concerned with commutants of \(0^*\)-algebras and makes the transition with the second part.

This second part is concerned with *-representations of *-algebras by unbounded operators. After some general concept developed in Chapter 8, Chapter 9 and 10 specialize to integrable representations of commutative *-algebras and of enveloping algebras. In Chapter 11 n-positive and completely positive *-representations are considered while Chapter 12 is concerned with integral decomposition of *-representations and states.

Each chapter ends with notes and bibliographical comments, while an up to date bibliography closes the book.

Reviewer: G.Loupias

### MSC:

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

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