K-Theory for Operator Algebras (Mathematical Sciences Research Institute Publications, Series Number 5)

K -Theory has revolutionized the study of operator algebras in the last few years. As the primary component of the subject of "noncommutative topol­ ogy," K -theory has opened vast new vistas within the structure theory of C\*­ algebras, as well as leading to profound and unexpected applications of opera­ tor algebras to problems in geometry and topology. As a result, many topolo­ gists and operator algebraists have feverishly begun trying to learn each others' subjects, and it appears certain that these two branches of mathematics have become deeply and permanently intertwined. Despite the fact that the whole subject is only about a decade old, operator K -theory has now reached a state of relative stability. While there will undoubtedly be many more revolutionary developments and applications in the future, it appears the basic theory has more or less reached a "final form." But because of the newness of the theory, there has so far been no comprehensive treatment of the subject. It is the ambitious goal of these notes to fill this gap. We will develop the K -theory of Banach algebras, the theory of extensions of C\*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects. We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text. K -Theory has revolutionized the study of operator algebras in the last few years. As the primary component of the subject of "noncommutative topolƯ ogy," K -theory has opened vast new vistas within the structure theory of C*Ư algebras, as well as leading to profound and unexpected applications of operaƯ tor algebras to problems in geometry and topology. As a result, many topoloƯ gists and operator algebraists have feverishly begun trying to learn each others' subjects, and it appears certain that these two branches of mathematics have become deeply and permanently intertwined. Despite the fact that the whole subject is only about a decade old, operator K -theory has now reached a state of relative stability. While there will undoubtedly be many more revolutionary developments and applications in the future, it appears the basic theory has more or less reached a "final form." But because of the newness of the theory, there has so far been no comprehensive treatment of the subject. It is the ambitious goal of these notes to fill this gap. We will develop the K -theory of Banach algebras, the theory of extensions of C*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects. We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text K-theory has helped convert the theory of operator algebras from a simple branch of functional analysis to a subject with broad applicability throughout mathematics, especially in geometry and topology, and many mathematicians of diverse backgrounds must learn the essential parts of the theory. This book is the only comprehensive treatment of K-theory for operator algebras, and is intended to help students, non specialists, and specialists learn the subject. This first paperback printing has been revised and expanded and contains an updated reference list. This book develops K-theory, the theory of extensions, and Kasparov's bivariant KK-theory for C*-algebras. Special topics covered include the theory of AF algebras, axiomatic K-theory, the Universal Coefficient Theorem, and E-theory. Although the book is technically complete, motivation and intuition are emphasized. Many examples and applications are discussed. K-theory, originally the study of vector bundles in topology, has become a powerful tool in the subject of operator algebras and has led to profound and unexpected applications throughout mathematics. This book develops K-theory and its deep bivariant version, Kasparov's KK-theory, from scratch to its most advanced aspects and describes important applications in topology and geometry. Numerous exercises ranging from elementary to research level supplement the text. Front Matter....Pages i-ix Introduction to K-Theory....Pages 1-17 Preliminaries....Pages 18-30 K 0 -Theory and Order....Pages 31-65 K 1 —Theory and Bott Periodicity....Pages 66-80 K-Theory of Crossed Products....Pages 81-118 More Preliminaries....Pages 119-142 Theory of Extensions....Pages 143-170 Kasparov’s KK-Theory....Pages 171-265 Further Topics....Pages 266-319 Back Matter....Pages 320-338