Customer Reviews

An Introduction to K-Theory for C*-Algebras (London Mathematical Society Student Texts)

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It's a very clear book with virtually no typos/mistakes and lots of nice exercises (if you are willing to do them - highly recommended). Really an introductory text, so you might need to consult the more advanced books of Wegge-Olsen, Blackadar after this one for more information on the subject. Nice place to start, especially if it is for self-study.

K-theory is a branch of algebraic topology originally concerned with the study of vector bundles by algebraic means. The first notions of the theory were put forward by Alexander Groethendieck in his work on the Riemann-Roch theorem in algebraic geometry, and early in the 60's it was developed into a branch of algebraic topology by M. Atiyah and F. Hirzebruch. From the analysis perspective, K-theory has a very natural link with the theory of Fredholm operators on a compact manifold and hence to the famous Atiyah-Singer index theorem. In the recent decades the theory has revolutionized the study of the structure theory of certain operator algebras. The procedure involves defining a collection of functors {K_n} from the category of C*-algebras to the category of abelian groups, satisfying the Eilenberg-Steenrod axioms for a homology theory. Bott periodicity as a handy feature then implies that there are only two such functors. In this nice text, the readers will find the needed material on C*-algebras, as well as an exposition of the K_0 and K_1 functors leading to the exploration of the Bott periodicity theorem and the six term exact sequence. As a graduate student a few years ago I attempted giving a seminar talk on this topic but was overwhelmed with the task of fitting the needed discussion into a 60-minute time span, specially in a way that the majority of the attendees could follow on. The exposition is indeed heavily algebraic in nature and hence anyone attempting to read and digest it properly will have to possess a strong background in the ideas and methods of basic algebraic topology.