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Book DetailsFields Institute MonographsVolume: 6; 1996; 309 ppMSC: Primary 46; Secondary 16; 41; 47
The subject of C*algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*algebras and Elliott's use of \(K\)theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*algebras.
This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topics include AF algebras, Bunce–Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*algebras, discrete crossed products, abelian C*algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions.
These notes were compiled during the author's participation in the special year on C*algebras at The Fields Institute for Research in Mathematical Sciences during the 1994–1995 academic year. The field of C*algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, \(K\)theory, and topology. The variety of examples offered in this text expose the student to many of these connections. Graduate students with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably selfcontained, and the author has provided results from other areas when necessary.
Customers in India, Sri Lanka, Bangladesh, and Pakistan, please contact Hindustan Book Agency (India), 17 U B Jawahar Nagar, Delhi 110 007, India.
Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipGraduate students, research mathematicians and physicists interested in functional analysis, C*algebras, and operator theory.

Table of Contents

Chapters

Chapter I. The basics of C*algebras

Chapter II. Normal operators and abelian C*algebras

Chapter III. Approximately finite dimensional (AF) C*algebras

Chapter IV. $K$theory for AF C*algebras

Chapter V. C*algebras of isometries

Chapter VI. Irrational rotation algebras

Chapter VII. Group C*algebras

Chapter VIII. Discrete crossed products

Chapter IX. BrownDouglasFillmore theory


Reviews

The book ... occupies a special place among numerous textbooks and monographs as in the subject chosen by the author so also in the method of presentation of the material ... Each chapter contains numerous interesting exercises ... can be recommended as a good introduction in the theory of \(C^*\)theory for all who are not yet under the charm of this remarkable theory as well as for specialists in the field as a collection of useful and carefully selected tools and methods ... useful for numerous lectures as a textbook for special courses with different aspects of \(C^*\)algebras.
Zentralblatt MATH 
Altogether, a very commendable introduction and collection.
Monatshefte für Mathematik 
One can assign parts of Davidson's book to good students learning the subject and expect good results ... the details persist even when the going gets tough ... this is the only book I know in which one can go through the BDF classification of essentially normal operators and follow, point by point, to the end.
Bulletin of the AMS 
The writing is exceptionally clear and easy to follow ... an outstanding book that should be on every operator algebraist's bookshelf.
Mathematical Reviews 
Very useful as a compendium of the common classes of nuclear \(C^*\)algebras, to which it provides an excellent introduction. A good collection of exercises is provided.
Bulletin of the London Mathematical Society 
A great book for a beginning grad course!
Palle Jorgensen


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The subject of C*algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*algebras and Elliott's use of \(K\)theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*algebras.
This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topics include AF algebras, Bunce–Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*algebras, discrete crossed products, abelian C*algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions.
These notes were compiled during the author's participation in the special year on C*algebras at The Fields Institute for Research in Mathematical Sciences during the 1994–1995 academic year. The field of C*algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, \(K\)theory, and topology. The variety of examples offered in this text expose the student to many of these connections. Graduate students with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably selfcontained, and the author has provided results from other areas when necessary.
Customers in India, Sri Lanka, Bangladesh, and Pakistan, please contact Hindustan Book Agency (India), 17 U B Jawahar Nagar, Delhi 110 007, India.
Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Graduate students, research mathematicians and physicists interested in functional analysis, C*algebras, and operator theory.

Chapters

Chapter I. The basics of C*algebras

Chapter II. Normal operators and abelian C*algebras

Chapter III. Approximately finite dimensional (AF) C*algebras

Chapter IV. $K$theory for AF C*algebras

Chapter V. C*algebras of isometries

Chapter VI. Irrational rotation algebras

Chapter VII. Group C*algebras

Chapter VIII. Discrete crossed products

Chapter IX. BrownDouglasFillmore theory

The book ... occupies a special place among numerous textbooks and monographs as in the subject chosen by the author so also in the method of presentation of the material ... Each chapter contains numerous interesting exercises ... can be recommended as a good introduction in the theory of \(C^*\)theory for all who are not yet under the charm of this remarkable theory as well as for specialists in the field as a collection of useful and carefully selected tools and methods ... useful for numerous lectures as a textbook for special courses with different aspects of \(C^*\)algebras.
Zentralblatt MATH 
Altogether, a very commendable introduction and collection.
Monatshefte für Mathematik 
One can assign parts of Davidson's book to good students learning the subject and expect good results ... the details persist even when the going gets tough ... this is the only book I know in which one can go through the BDF classification of essentially normal operators and follow, point by point, to the end.
Bulletin of the AMS 
The writing is exceptionally clear and easy to follow ... an outstanding book that should be on every operator algebraist's bookshelf.
Mathematical Reviews 
Very useful as a compendium of the common classes of nuclear \(C^*\)algebras, to which it provides an excellent introduction. A good collection of exercises is provided.
Bulletin of the London Mathematical Society 
A great book for a beginning grad course!
Palle Jorgensen