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An introduction to $$K$$-theory for $$C^*$$-algebras. (English) Zbl 0967.19001
London Mathematical Society Student Texts. 49. Cambridge: Cambridge University Press. xii, 242 p. (2000).
This elementary introduction to the $$K$$-theory of $$C^\ast$$-algebras is based on lectures given by the first named author and elaborated by his two coauthors. The subject matter is chosen and arranged to be covered in a one-semester graduate course. The content is as follows: 1. $$C^\ast$$-algebra theory. 2. Projections and unitary elements. 3. The $$K_0$$-group of a unital $$C^\ast$$-algebra. 4. The functor $$K_0$$. 5. The ordered abelian group $$K_0(A)$$. 6. Inductive limit $$C^\ast$$-algebras. 7. Classification of AF-algebras. 8. The functor $$K_1$$. 9. The index map. 10. The higher $$K$$-functors. 11. Bott periodicity. 12. The six-term exact sequence. 13. Inductive limits of dimension drop algebras.
The repertoire of examples is restricted to a minimum, the more sophisticated examples such as Cuntz algebras and irrational rotation algebras being treated only sketchyly in the exercises. The emphasis lies on the classification of AF-algebras and on dimension drop algebras. The absence of crossed products makes it necessary to choose, e.g., Atiyah’s proof of the periodicity theorem via deformation of Laurent loops to linear ones. The textbook is a nice introduction to the subject preparing the ground for the study of more advanced texts such as the well-known ones by B. Blackadar or N. E. Wegge-Olsen.

##### MSC:
 19-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to $$K$$-theory 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 19Kxx $$K$$-theory and operator algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)