Theory of Operator Algebras III

Masamichi Takesaki (auth.)9783540429135, 3540429131

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.

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Table of contents :
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