Classification of Nuclear C*-Algebras. Entropy in Operator Algebras (Encyclopaedia of Mathematical Sciences (126))

This EMS volume consists of two parts, written by leading scientists in the field of operator algebras and non-commutative geometry. The first part, written by M.Rordam entitled "Classification of Nuclear, Simple C\*-Algebras" is on Elliotts classification program. The emphasis is on the classification by Kirchberg and Phillips of Kirchberg algebras: purely infinite, simple, nuclear separable C\*-algebras. This classification result is described almost with full proofs starting from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for exact C\*-algebras. The classificatin of finite simple C\*-algebras starting with AF-algebras, and continuing with AF- and AH-algberas) is covered, but mostly without proofs. The second part, written by E.Stormer entitled "A Survey of Noncommutative Dynamical Entropy" is a survey of the theory of noncommutative entropy of automorphisms of C\*-algebras and von Neumann algebras from its initiation by Connes and Stormer in 1975 till 2001. The main definitions and resuls are discussed and illustrated with the key examples in the theory. This book will be useful to graduate students and researchers in the field of operator algebras and related areas. 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. Afactor 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. This EMS volume consists of two parts, written by leading scientists in the field of operator algebras and non-commutative geometry. The first part, written by M. Rordam entitled "Classification of Nuclear, Simple C*-Algebras" is on Elliotts classification program. The emphasis is on the classification by Kirchberg and Phillips of Kirchberg algebras: purely infinite, simple, nuclear separable C*-algebras. This classification result is described almost with full proofs starting from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for exact C*-algebras. The classificatin of finite simple C*-algebras starting with AF-algebras, and continuing with AF- and AH-algberas) is covered, but mostly without proofs. The second part, written by E. Stormer entitled "A Survey of Noncommutative Dynamical Entropy" is a survey of the theory of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation by Connes and Stormer in 1975 till 2001. The main definitions and resuls are discussed and illustrated with the key examples in the theory. This book will be useful to graduate students and researchers in the field of operator algebras and related areas Part I. Classification of Nuclear, Simple C*-Algebras, M.Rordam: 1. AF-algebras and their Classification 2. Preliminaries 3. Classification results for finite C*-algebras 4. Purely infinite simple C*-algebras 5. On O 2 6. Nuclear and exact C*-algebras and exact C*-algebras 7. Tensor products by O 2 and O Öinfty 8. Classification of Kirchberg algebras Part II. A Survey of Noncommutative Dynamical Entropy, E. Stormer: Introduction 1. Entropy in finite von Neumann algebras 2. Entropy in C*-algebras 3. Bogoliubov automorphisms 4. The entropy of Sauvageot and Thouvenot 5. Voiculescu's approximation entropies 6. Crossed products 7. Free products 8. Binary shifts 9. Generators 10. The variational principle. Front Matter....Pages I-IX Classification of Nuclear, Simple C *-algebras....Pages 1-145 A Survey of Noncommutative Dynamical Entropy....Pages 147-198