Theory of Operator Algebras III (Encyclopaedia of Mathematical Sciences (127))

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, II 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. Together with "Theory of Operator Algebras I, II" (EMS 124 and 125), this book, written by one of the most prominent researchers in the field of operator algebras, presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis. It is is part of the recently developed part of the "Encyclopaedia of Mathematical Sciences" on operator algebras and non-commutative geometry (see http://www.springer.de/math/ems/index.html). The book provides essential and comprehensive information for graduate students and researchers in mathematics and mathematical physics. Front Matter....Pages I-XXII Ergodic Transformation Groups and the Associated von Neumann Algebras....Pages 1-80 Approximately Finite Dimensional von Neumann Algebras....Pages 81-152 Nuclear C *-Algebras....Pages 153-204 Injective von Neumann Algebras....Pages 205-251 Non-Commutative Ergodic Theory....Pages 252-295 Structure of Approximately Finite Dimensional Factors....Pages 296-411 Subfactors of an Approximately Finite Dimensional Factor of Type II 1 ....Pages 412-493 Back Matter....Pages 495-548 Preface Chapter XIII. Ergodic Transformaton Groups and the Associated von Neumann Algebras Chapter XIV. Approximately Finite Dimensional von Neumann Algebras Chapter XV. Nuclear C*-Algebras Chapter XVI. Injective von Neumann Algebras Chapter XVII. Non-Commutative Ergodic Theory Chapter XVIII. Structure of Approximately Finite Dimensional Factors Chapter XIX. Subfactors of a Factor of Tape II1 Appendix Bibliography Index.

From the reviews: "These three bulky volumes [EMS 124, 125, 127] [...] provide an introduction to this rapidly developing theory. [...] These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of mathematics. Furthermore, they should be on the bookshelf of every researcher of the area." Acta Scientiarum Mathematicarum

This publication, written by one of the most prominent researchers in the field of operator algebras, summarises the scientific work of the author focusing on von Neumann algebras and non-commutative integration M. Takesaki. Includes Bibliographcial References And Indexes.