\(K\)-theory and \(C^*\)-algebras: a friendly approach.

*(English)*Zbl 0780.46038
Oxford Science Publications. Oxford: Oxford University Press. xii, 370 p. (1993).

\(K\)-theory has revolutionized the study of operator algebras in the last few years. This book is a brilliantly written introduction to the \(K\)- theory of \(C^*\)-algebras. It is divided into 4 parts.

The first part (Chapters 1-5) is devoted to \(C^*\)-algebras. Chapter 1 lists all the assumed \(C^*\)-algebraic prerequisites and proves a few results about stabilization. The multiplier algebra is constructed in Chapter 2. In Chapter 3 the methods for classifying exact sequences of \(C^*\)-algebras is established. Chapter 4 is concerned with the properties of invertible and unitary elements in \(C^*\)-algebras and Chapter 5 is concerned with the equivalence theory for projections.

Part 2 (Chapters 6-13) of the book may be considered as its central part. Basic properties of the \(K_ *\)-functor are proved in Chapters 6, 7, 9, 11, 13. The index map in \(K\)-theory is constructed in Chapter 8. Chapter 10 is devoted to a useful result: \(K_ *\)-functors for multiplier algebras of stable \(C^*\)-algebras are trivial. Three concrete classes of \(C^*\)-algebras are discussed in Chapter 12.

Part 3 (Chapters 14-17) can be viewed as an illustration of general \(K\)- theory on a certain generalized Fredholm index theory. Here the author follows Mingo in the concrete operator realization of a \(K\)-theoretical index map that resembles the Fredholm index very much. The appendices in Part 4 extend Part 1 in establishing important \(C^*\)-algebraic technical remedies of more general character.

The first part (Chapters 1-5) is devoted to \(C^*\)-algebras. Chapter 1 lists all the assumed \(C^*\)-algebraic prerequisites and proves a few results about stabilization. The multiplier algebra is constructed in Chapter 2. In Chapter 3 the methods for classifying exact sequences of \(C^*\)-algebras is established. Chapter 4 is concerned with the properties of invertible and unitary elements in \(C^*\)-algebras and Chapter 5 is concerned with the equivalence theory for projections.

Part 2 (Chapters 6-13) of the book may be considered as its central part. Basic properties of the \(K_ *\)-functor are proved in Chapters 6, 7, 9, 11, 13. The index map in \(K\)-theory is constructed in Chapter 8. Chapter 10 is devoted to a useful result: \(K_ *\)-functors for multiplier algebras of stable \(C^*\)-algebras are trivial. Three concrete classes of \(C^*\)-algebras are discussed in Chapter 12.

Part 3 (Chapters 14-17) can be viewed as an illustration of general \(K\)- theory on a certain generalized Fredholm index theory. Here the author follows Mingo in the concrete operator realization of a \(K\)-theoretical index map that resembles the Fredholm index very much. The appendices in Part 4 extend Part 1 in establishing important \(C^*\)-algebraic technical remedies of more general character.

Reviewer: V.Deundyak (Rostov-na-Donu)

##### MSC:

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

19K14 | \(K_0\) as an ordered group, traces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |