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K-theory for operator algebras. (English) Zbl 0597.46072
Mathematical Sciences Research Institute Publications, 5. New York etc.: Springer-Verlag. VII, 338 p. DM 78.00 (1986).
In the last ten years major advances in operator algebras came from K- theory. This is based on the fact that knowledge of the similarity of projections or the connectivity of the group of invertibles in the endomorphism algebras of finitely generated projective modules over a \(C^*\)-algebra gives much insight into the structure of the \(C^*\)- algebra itself. Specifically this lead to the classification of AF algebras via dimension groups and helped to understand group \(C^*\)- algebras and crossed product algebras. Starting with the Gelfand-Naimark theorem arbitrary \(C^*\)-algebras can be looked at as ”non-commutative topological spaces” with operator K-theory as the corresponding K- (cohomology) theory. K-theory and the dual K-homology became then incorporated in Kasparov’s KK-theory which is now the natural framework for the index theory of elliptic pseudodifferential operators. The monograph under review is the first comprehensive introduction to this field. The reader is supposed to have a good background in operator theory (since a possible title of the book could also have been ”K-theory for operator algebraists”).
The first half of the book, chapters I to V, presents operator K-theory and its applications to C*-algebras mentioned above. The second half, chapters VI to VIII, gives a readable introduction to KK-theory which is more accessible than Kasparov’s original papers since most of the recent improvements and simplifications are incorporated. The last chapter gives diverse further topics, especially there is a section on applications to geometry and topology, but here (as at some places before) the reader is urged to consult the original papers for details. Most of the sections include exercises and problems. The exercises range from easy to difficult ones with references to the literature. The problems are sometimes challenging with suggestions to future research.
The book ends with an excellent bibliography of more than 200 items, and a subject index.
Reviewer: H.Strich

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L05 General theory of \(C^*\)-algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46L55 Noncommutative dynamical systems
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)