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

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.

Table of Contents:
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.

Review :
From the reviews: "... These notes [by E.Stormer] describe the main approaches to noncommutative entropy, together with several ramifications and variants. The notion of generator and variational principle are used to give applications to subfactors and C*-algebra formalism of quantum statistical mechanics. The author considers the most frequently studied examples, including Bernoulli shifts, Bogolyubov automorphisms, dual automorphisms on crossed products, shifts on infinite free products, and binary shifts on the CAR-algebra. The mathematical techniques and ideas are beautifully exposed, and the whole paper is a rich resource on the subject, either for the expert or the beginner. ..." V.Deaconu, Mathematical Reviews 2004 "... the author gives a clear presentation of the dramatic developments in the classification theory for simple C*-algebras that have taken place over the past 25 years or so. ... As there is such a large amount of literature on the subject, this monograph article is particularly useful to the relative novice who wants to know the fundamental results in the theory without wading through a massive amount of detail. ...This monograph-length article is extremely well-written, filled with concrete examples, and has an exhaustive bibliography. I recommend it as an excellent introduction to graduate students and other mathematicians who want to bring themselves up-to-date on the subject. .." J.A.Packer, Mathematical Reviews 2004 "Both contributions to this volume are high-end, excellently written research reviews, reflecting very thoroughly the current status in the respectively treated subbranches of the quickly evolving complex field of C* algebra theory. They both give a beautiful lay-out of the vast research program in the field which has been going on for decades ... as well as to the standard works. ... an excellent, very thorough, concise and needed overview for the researcher who is active in this field." (Mark Sioen, Bulletin of the Belgian Mathematical Society, 2007)