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Book DetailsGraduate Studies in MathematicsVolume: 16; 1997; 676 ppMSC: Primary 46; Secondary 47The return of a classic!
This work and Fundamentals of the Theory of Operator Algebras. Volume I, Elementary Theory present an introduction to functional analysis and the initial fundamentals of \(C^*\) and von Neumann algebra theory in a form suitable for both intermediate graduate courses and selfstudy. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is selfcontained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences.
Praise for both volumes ...
“ ...these two volumes represent a magnificent achievement. They will be an essential item on every operator algebraist's bookshelves and will surely become the primary source of instruction for research students in von Neumann algebra theory.”
—Bulletin of the London Mathematical Society
“Volumes I and II were published in 1982 and 1983. Since then they have quickly established themselves as The Textbooks in operator algebra theory.”
—Bulletin of the American Mathematical Society
“One of the splendid features of the original two volumes is their large supply of exercises ... which illustrate the results of the text and expand its scope.”
—L'Enseignement mathématique
ReadershipGraduate students, research mathematicians, educators, and mathematical physicists interested in functional analysis, operator algebras, and applications.

Table of Contents

Chapters

Chapter 6. Comparison theory of projections

Chapter 7. Normal states and unitary equivalence of von Neumann algebras

Chapter 8. The trace

Chapter 9. Algebra and commutant

Chapter 10. Special representations of $C^*$algebras

Chapter 11. Tensor products

Chapter 12. Approximation by matrix algebras

Chapter 13. Crossed products

Chapter 14. Direct integrals and decompositions


Reviews

What is presented here is an exciting and careful account of some very substantial mathematics! ... As with the first volume, this book has an outstanding collection of exercises (and groupings of exercises) ... Although the material in this second volume is substantially more advanced than that in the first volume, the authors have maintained the same spirit of freshness of the presentation given there.
Mathematical Reviews 
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.
Zentralblatt MATH 
This volume by one of the cofounders of the isomorphism theory of Bernoulli processes offers a comprehensive treatment of finitestate stationary processes, with special emphasis on their combinatorial and coding properties. It is of interest to advanced graduate students and researchers working in ergodic theory, probability and information and gives an excellent account of many different tools and constructions of measuretheoretic dynamics ... There is a useful bibliography which allows the interested reader to trace the history of the ideas presented in the book.
Monatshefte für Mathematik


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This work and Fundamentals of the Theory of Operator Algebras. Volume I, Elementary Theory present an introduction to functional analysis and the initial fundamentals of \(C^*\) and von Neumann algebra theory in a form suitable for both intermediate graduate courses and selfstudy. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is selfcontained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences.
Praise for both volumes ...
“ ...these two volumes represent a magnificent achievement. They will be an essential item on every operator algebraist's bookshelves and will surely become the primary source of instruction for research students in von Neumann algebra theory.”
—Bulletin of the London Mathematical Society
“Volumes I and II were published in 1982 and 1983. Since then they have quickly established themselves as The Textbooks in operator algebra theory.”
—Bulletin of the American Mathematical Society
“One of the splendid features of the original two volumes is their large supply of exercises ... which illustrate the results of the text and expand its scope.”
—L'Enseignement mathématique
Graduate students, research mathematicians, educators, and mathematical physicists interested in functional analysis, operator algebras, and applications.

Chapters

Chapter 6. Comparison theory of projections

Chapter 7. Normal states and unitary equivalence of von Neumann algebras

Chapter 8. The trace

Chapter 9. Algebra and commutant

Chapter 10. Special representations of $C^*$algebras

Chapter 11. Tensor products

Chapter 12. Approximation by matrix algebras

Chapter 13. Crossed products

Chapter 14. Direct integrals and decompositions

What is presented here is an exciting and careful account of some very substantial mathematics! ... As with the first volume, this book has an outstanding collection of exercises (and groupings of exercises) ... Although the material in this second volume is substantially more advanced than that in the first volume, the authors have maintained the same spirit of freshness of the presentation given there.
Mathematical Reviews 
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.
Zentralblatt MATH 
This volume by one of the cofounders of the isomorphism theory of Bernoulli processes offers a comprehensive treatment of finitestate stationary processes, with special emphasis on their combinatorial and coding properties. It is of interest to advanced graduate students and researchers working in ergodic theory, probability and information and gives an excellent account of many different tools and constructions of measuretheoretic dynamics ... There is a useful bibliography which allows the interested reader to trace the history of the ideas presented in the book.
Monatshefte für Mathematik