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Fundamentals of the theory of operator algebras. Volume II: Advanced theory. (English) Zbl 0601.46054

This book, written by two of the main contributors to the subject, is the first introduction to operator algebras that aims at being pedagogical. It concentrates on the classical non-commutative measure theory aspect of operator algebras (as opposed to ”non-commutative topology” and ”differential geometry” which have been developed much more recently), thus on von Neumann algebras - and on representation theory as far as \(C^*\)-algebras are concerned. Even with this restriction the theory has by now grown so complicated that it seems to be impossible to give more than the fundamentals of the theory in a pedagogical text. The authors do a very fine job in exposing these fundamentals up to and including Tomita-Takesaki theory.
The second volume, under review (for part I see Zbl 0518.46046), starts with chapter 6 on Murray-von Neumann comparison theory for projections which remains up to this day one of the most beautiful and fertile ideas in von Neumann-algebra theory, allowing to associate a real valued dimension to certain subspaces of a Hilbert space.
The following chapter 7 develops the theory of normal states and contains a nice proof of the result that algebraic isomorphisms between von Neumann algebras with cyclic and separating vectors are implemented by unitaries. Chapter 8 is devoted to the existence of the trace which is the ”integral” associated to the ”measure” given by the dimension function alluded to above. It also contains the construction of some further examples of factors (others had been constructed in chapter 6).
Chapter 9 contains the study of the close connections that exist between a von Neumann algebra and its commutant and of modular (Tomita-Takesaki) theory. This theory is the basis for all of the more recent work in the classification of von Neumann algebras (Connes, Haagerup etc...) and is explained here with the care that it deserves. The necessary prerequisits on unbounded operators are contained in volume I. Chapter 10 contains some material on representations of \(C^*\) algebras derived from von Neumann algebra theory. Chapter 11 treats (finite and infinite) tensor products of \(C^*\)- and von Neumann algebras in a standard way.
Chapter 12 then studies the very important examples of \(C^*\)- and von Neumann algebras obtained as infinite tensor products of matrix algebras and their classification. As is well known, in the \(C^*\)-algebra case the isomorphism class of an infinite tensor product \(\otimes^{\infty}_{k=1}M_{n_ k}\) (also called an UHF-algebra), with \(M_{n_ k}\) the \(n_ k\times n_ k\) complex matrices, depends on a certain invariant associated with the numbers \(\{n_ k\}\), while in the von Neumann algebra case one obtains independently of the choice of the \(\{n_ k\}\) the ”hyperfinite” or ”matricial” factor of type II. The Powers construction of a continuum of pairwise non-isomorphic factors of type III, based on representations of an UHF-algebra is also given.
Chapter 13 studies discrete and continuous crossed products of von Neumann algebras, and the particular case of the crossed product of a type III factor by its modular automorphism group. The last chapter 14 finally contains the theory of the decomposition of a general von Neumann algebra as a direct integral (a generalization of the direct sum) of factors (a particular case is the disintegration of a unitary representation of a locally compact group). This theory permits to reduce most questions about von Neumann algebras to questions about factors.
Globally, this book is extremely clear and well written and ideally suited for an introductory course on the subject or for a student who wishes to learn the fundamentals of the classical theory of operator algebras. The authors plan two further volumes containing solutions to the many exercises in the text.
Reviewer: J.Cuntz

MSC:

46L10 General theory of von Neumann algebras
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L35 Classifications of \(C^*\)-algebras
46L45 Decomposition theory for \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras

Citations:

Zbl 0518.46046