##
**Theory of operator algebras. II.**
*(English)*
Zbl 1059.46031

Encyclopaedia of Mathematical Sciences 125. Operator Algebras and Non-commutative Geometry 6. Berlin: Springer (ISBN 3-540-42914-X/hbk). xxii, 518 p. (2003).

The fundamental work on the theory of operator algebras was first started by John von Neumann and F. J. Murray in the 1930’s. By an operator algebra we mean a closed, not necessarily self-adjoint, subalgebra of \(B(H)\) for some Hilbert space \(H\). Operator algebras are, in a certain sense, a generalization of the algebra of complex matrices.

This is the second volume of an advanced textbook written by one of the most active researchers in the theory of operator algebras and is useful for graduate students and specialists.

The first volume is an account of similarity between the theory of measures on a locally compact space and the theory of operator algebras and may be regarded as a non-commutative integration, cf. M. Takesaki [Theory of operator algebras I. 2nd printing of the 1979 ed. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 124(5). Berlin: Springer)(2002; Zbl 0990.46034)]. The third volume is focused on structure analysis of approximately finite dimensional factors and their automorphism groups, cf. M. Takesaki [Theory of operator algebras III. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 127(VIII). Berlin: Springer)(2003; Zbl 1059.46032)].

In the second volume which is well organized, one can find an investigation of structure of von Neumann algebras of type III and their automorphism groups and consists of seven chapters (6-12), notation index, and subject index, and also jointly with vol. III consists of an author preface, an appendix containing twenty five sections and a bibliography containing 334 references. Each chapter begins with a clear introduction describing the content of that chapter, contains several interesting exercises and is concluded with a section of rich historical notes. There are referred to some publications listing merely in the bibliography of the first volume.

In the following paragraphs we briefly describe the chapters of the book:

Chapter VI. Left Hilbert Algebras (a foundation of non-commutative integration). This chapter is devoted to investigation of direct integral of left Hilbert algebras and Tomita algebras.

Chapter VII. Weights (corresponding to measures on a space equipped with a \(\sigma\)-algebra). In this chapter the correspondence of a faithful semi-finite normal weight and a full left Hilbert algebra is given, in particular Plancherel weight is introduced.

Chapter VIII. Modular Automorphism Groups (of a faithful semi-finite normal weight on a von Neumann algebra). This chapter contains an extension of Radon-Nikodym theorem to the non-commutative setting, and a complete characterization of the semi-finiteness in terms of the innerness of modular automorphism groups.

Chapter IX. Non-Commutative Integration (or theory of weights, traces and states). A concept of relative tensor product of a pair of a right module and a left module over a fixed von Neumann algebra is introduced in this chapter, as well as a non-commutative analogue of the notion of conditional expectation is considered.

Chapter X. Crossed Products and Duality (a systematic study of automorphism actions of a locally compact group on a von Neumann algebra). This chapter deals with a duality theorem for crossed products by abelian locally compact groups, playing a key role in the structure analysis of a von Neumann algebra of type III.

Chapter XI. Abelian Automorphism Group ( the spectral analysis of automorphism actions of locally compact abelian groups on von Neumann algebras). This chapter is devoted to a formal definition of the spectrum of an element of a von Neumann algebra relative to an action \(\alpha\) of a locally compact abelian group, and also the Connes spectrum of \(\alpha\) is introduced. In the rest of this chapter it is constructively proved that every derivation of a von Neumann algebra is inner (cf. R. V. Kadison [Ann. Math. (2) 83, 280-293 (1966; Zbl 0139.30503)] and S. Sakai [Ann. Math. (2) 83, 273-279 (1966; Zbl 0139.30601)]), and a characterization of inner automorphisms and the cohomology vanishing theorem are included.

Chapter XII. Structure of a von Neumann Algebra of Type III ( a general description). It is remarkable and mysterious that the most of von Neumann algebras appearing in quantum physics are of type III, cf. H. Araki and E. J. Woods [J. Mathematical Phys. 4 (1963), 637-662]. R. T. Powers [Ann. Math. (2) 86, 138-171 (1967; Zbl 0157.20605)] showed the existence of continuously many non-isomorphic factors of type III.

In this chapter, the structure factors of type III\(_{\lambda}, 0<\lambda<1\) and III\(_{0}\) is considered. The canonical construction of a non-commutative flow of weights of a von Neumann algebra is also discussed.

This is the second volume of an advanced textbook written by one of the most active researchers in the theory of operator algebras and is useful for graduate students and specialists.

The first volume is an account of similarity between the theory of measures on a locally compact space and the theory of operator algebras and may be regarded as a non-commutative integration, cf. M. Takesaki [Theory of operator algebras I. 2nd printing of the 1979 ed. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 124(5). Berlin: Springer)(2002; Zbl 0990.46034)]. The third volume is focused on structure analysis of approximately finite dimensional factors and their automorphism groups, cf. M. Takesaki [Theory of operator algebras III. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 127(VIII). Berlin: Springer)(2003; Zbl 1059.46032)].

In the second volume which is well organized, one can find an investigation of structure of von Neumann algebras of type III and their automorphism groups and consists of seven chapters (6-12), notation index, and subject index, and also jointly with vol. III consists of an author preface, an appendix containing twenty five sections and a bibliography containing 334 references. Each chapter begins with a clear introduction describing the content of that chapter, contains several interesting exercises and is concluded with a section of rich historical notes. There are referred to some publications listing merely in the bibliography of the first volume.

In the following paragraphs we briefly describe the chapters of the book:

Chapter VI. Left Hilbert Algebras (a foundation of non-commutative integration). This chapter is devoted to investigation of direct integral of left Hilbert algebras and Tomita algebras.

Chapter VII. Weights (corresponding to measures on a space equipped with a \(\sigma\)-algebra). In this chapter the correspondence of a faithful semi-finite normal weight and a full left Hilbert algebra is given, in particular Plancherel weight is introduced.

Chapter VIII. Modular Automorphism Groups (of a faithful semi-finite normal weight on a von Neumann algebra). This chapter contains an extension of Radon-Nikodym theorem to the non-commutative setting, and a complete characterization of the semi-finiteness in terms of the innerness of modular automorphism groups.

Chapter IX. Non-Commutative Integration (or theory of weights, traces and states). A concept of relative tensor product of a pair of a right module and a left module over a fixed von Neumann algebra is introduced in this chapter, as well as a non-commutative analogue of the notion of conditional expectation is considered.

Chapter X. Crossed Products and Duality (a systematic study of automorphism actions of a locally compact group on a von Neumann algebra). This chapter deals with a duality theorem for crossed products by abelian locally compact groups, playing a key role in the structure analysis of a von Neumann algebra of type III.

Chapter XI. Abelian Automorphism Group ( the spectral analysis of automorphism actions of locally compact abelian groups on von Neumann algebras). This chapter is devoted to a formal definition of the spectrum of an element of a von Neumann algebra relative to an action \(\alpha\) of a locally compact abelian group, and also the Connes spectrum of \(\alpha\) is introduced. In the rest of this chapter it is constructively proved that every derivation of a von Neumann algebra is inner (cf. R. V. Kadison [Ann. Math. (2) 83, 280-293 (1966; Zbl 0139.30503)] and S. Sakai [Ann. Math. (2) 83, 273-279 (1966; Zbl 0139.30601)]), and a characterization of inner automorphisms and the cohomology vanishing theorem are included.

Chapter XII. Structure of a von Neumann Algebra of Type III ( a general description). It is remarkable and mysterious that the most of von Neumann algebras appearing in quantum physics are of type III, cf. H. Araki and E. J. Woods [J. Mathematical Phys. 4 (1963), 637-662]. R. T. Powers [Ann. Math. (2) 86, 138-171 (1967; Zbl 0157.20605)] showed the existence of continuously many non-isomorphic factors of type III.

In this chapter, the structure factors of type III\(_{\lambda}, 0<\lambda<1\) and III\(_{0}\) is considered. The canonical construction of a non-commutative flow of weights of a von Neumann algebra is also discussed.

Reviewer: Mohammad Sal Moslehian (Mashhad)

### MSC:

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

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

46L10 | General theory of von Neumann algebras |