to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, IT and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.
Rezensionen / Stimmen
From the reviews:
"... These three bulky volumes [EMS 124, 125, 127], written by one of the most prominent researchers of the area, provide an introduction to this repidly developing theory. ... These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of matematics. Furthermore, they should be on the bookshelf of every researcher of the area."
László Kérchy, Acta Scientiarum Mathematicarum, Vol. 69, 2003
"... The author has obviously put a tremendous amount of work into the writing of this book and he is to be congratulated for making it available to the operaotr algebra community. ..."
R.S.Doran, Mathematical Reviews Clippings from Issue 2004g
"Theory of Operator Algebras II" von Masmichi Takesaki ist der zweite Band eines dreibändigen Werks und behandelt neuere Entwicklungen in der Theorie der Operatoralgebren der letzten 20 Jahre. ......
Die Auswahl der Themen ist in erheblichem Maß durch die Entwicklung der nichtkommutativen Geometrie geprägt. Daher stellt das Buch eine äußerst nützliche Sammlung von theoretischem Hintergrundwissen dar, das für das Verständnis von Originalarbeiten der nichtkommutativen Geometrie notwendig ist, und in dieser kompakten Form bisher nicht zu finden war.
Geschrieben in einem schnörkellosen, aber klaren, didaktischen Stil ist dieses gut strukturierte Buch jedem zu empfehlen, der sich ernsthaft mit den Grundlagen der Operatoralgebra und der nichtkommutativen Geometrie auseinandersetzen will. ..."
Christian Blohman, for complete review see http://www.wissenschaft-online.de/artikel/714368
"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. . Each chapter begins with a clear introduction describing the content of thatchapter, contains several interesting exercises and is concluded with a section of rich historical notes." (Mohammad Sal Moslehian, Zentralblatt MATH, Vol. 1059 (10), 2005)
"This book, written by one of the world's most respected operator algebraists, is devoted primarily to the study of type III von Neumann algebras . . It contains seven chapters and an extensive appendix . . each chapter has its own introduction describing the content and basic strategy, enabling the reader to get a quick overview of the results. . The author has obviously put a tremendous amount of work into the writing of this book and he is to be congratulated . ." (Robert S. Doran, Mathematical Reviews, 2004 g)
