Theory of operator algebras. III
As author stated ''the theory of operator algebras should be viewed as a number theory in analysis'' from the point of a historical or cultural view. This is the third volume of an a modern book written by one of main contributors to the theory of operator algebras and is useful for researchers. The first volume deals with the 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. \textit{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)]. In the second volume one can find an investigation of structure of von Neumann algebras of type III and their automorphism groups, cf. \textit{M. Takesaki} [Theory of operator algebras II. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 125(VI). Berlin: Springer)(2003; Zbl 1059.46031)]. The third volume is a comprehensive account of structure analysis of approximately finite dimensional factors and their automorphism groups and consists of seven chapters (13-19), notation index, and subject index, and also jointly with vol. II 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. Some materials in the theory of operator algebras such as the classification of nuclear \(C^{*}\)-algebras, free probability theory, entropy theory, and \(K\)-theory for \(C^{*}\)-algebras are not included in the book and Encyclopaedia subseries editors will plan to include them and other topics on operator algebras and non-commutative geometry in the other volumes. In the following paragraphs we briefly describe the chapters of the book: \textbf{Chapter XIII. Ergodic Transformation Groups and Associated von Neumann Algebras} (a systematic study). This chapter deals with group measure space construction of factors, amenable groups and characterization of AF measured groupoids by amenability, cf. \textit{A. Connes, J. Feldman} and \textit{B. Weiss} [Ergodic Theory Dyn. Syst. 1, 431-450 (1981; Zbl 0491.28018)]. \textbf{Chapter XIV. Approximately Finite Dimensional von Neumann Algebras}. This chapter is devoted to introduce a new class of factors so-called approximately finite dimensional (AFD) factors and to show the uniqueness of AF II\(_{1}\)-factors. In addition, infinite tensor products of von Neumann algebras (cf. \textit{E. Størmer} [Am. J. Math. 93, 810-818 (1971; Zbl 0222.46047)]) and closedness of the inner automorphisms of a factor in the automorphism group are studied. \textbf{Chapter XV. Nuclear \(C^{*}\)-Algebras} (or amenable \(C^{*}\)-algebras). A \(C^{*}\)-algebra \(A\) is called nuclear if injective \(C^{*}\)-cross norm and projective \(C^{*}\)-cross norm on \(A\otimes B\) coincide for every \(C^{*}\)-algebra \(B\) (cf. \textit{E. G. Effros} and \textit{E. C. Lance} [Adv. Math. 25, 1-34 (1977; Zbl 0372.46064)]). A von Neumann algebra \(M\) is said to be injective if every completely positive map of any self-adjoint closed subspace \(N\) of any unital \(C^{*}\)-algebra \(A\) containing the identity of \(A\) into \(M\) can be extended to a completely positive map of \(A\) into \(M\). In this chapter, the characterization of a nuclear \(C^{*}\)-algebra \(A\) by means of the injectivity of the universal enveloping von Neumann algebra \(A^{**}\) is proved. In addition, Grothendieck-Haagerup-Pisier inequality and some of its applications is discussed. \textbf{Chapter XVI. Injective von Neumann Algebras} ( a significant class of von Neumann algebras). This chapter is emphasized on the equivalence of the nuclearity and approximately finite dimensionality of a von Neumann algebra, and as well as approximate finiteness of a finite injective von Neumann algebra. \textbf{Chapter XVII. Non-Commutative Ergodic Theory}. In this chapter, Rokhlin's tower theorem in the classical ergodic theory is extended to the non-commutative settings. Further, the outer conjugacy classes of approximately inner automorphisms of strongly stable factors is completely determined. \textbf{Chapter XVIII. Structure of Approximately Finite Dimensional Factors} ( a highlight of the theory of von Neumann algebras). Although we have no complete classification of factors in the none type I ( it is well-known that a factor of type I\(_{n}\) is isomorphic to \(M_{n}({\mathbb C})\) for \(1\leq n<\infty\) and to \({\mathcal L}(l^{2})\) for \(n=\infty\)), the structure of AFD factors (in each type)is well described in this chapter. \textbf{Chapter XIX. Subfactors of an Approximately Finite Dimensional Factor of Type II\(_{\text 1}\)} (the last chapter of the book). Pairs \(N\subseteq M\) of AFD factors of type II\(_{\text 1}\) with index less than four are classified in this chapter. Recall that a subfactor is an inclusion of II\(_{\text 1}\)-factors \(N \subseteq M\) such that the dimension of \(M\) as a left \(N\)-Hilbert module, so-called the Jones index \([M\colon N]\) (cf. \textit{V. F. R. Jones} [Invent. Math. 72, no. 1, 1-25 (1983; Zbl 0508.46040)]), is finite.
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