Theory of Operator Algebras II (Encyclopaedia of Mathematical Sciences (125))

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.

Together with Theory of Operator Algebras I, III (EMS 124 and 127), 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 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.

Together with Theory of Operator Algebras I and III, this book presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis.

From the reviews: "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

Front Matter....Pages I-XXII Left Hilbert Algebras....Pages 1-39 Weights....Pages 40-90 Modular Automorphism Groups....Pages 91-140 Non-Commutative Integration....Pages 141-236 Crossed Products and Duality....Pages 237-310 Abelian Automorphism Group....Pages 311-362 Structure of a von Neumann Algebra of Type III....Pages 363-461 Back Matter....Pages 463-518 Preface Chapter VI. Left Hilbert Algebras Chapter VII. Weights Chapter VIII. Modular Automorphism Groups Chapter IX. Non-Commutative Integration Chapter X. Crossed Products and Duality Chapter XI. Abelian Automorphism Groups Chapter XII. Structure of a von Neumann Algebra of Type III Appendix Bibliography Index. This publication, written by one of the most prominent researchers in the field of operator algebras, summarises the scientific work of the author focussing on von Neumann algebras and non-communicative integration