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92 pages, 2004

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Contents: Preliminaries on C*-Algebras, Basic definitions, Short exact sequences,Spectral theory , Unital C*-Algebras 27, Projections and Unitaries 17 2.1 Homotopy for unitaries ,The K0-Group for Unital C*-Algebras The Grothendieck Construction,Functoriality,The universal property of K0 Homotopy invariance, Definition of the K0-Functor, 4.3 Inductive Limits. Continuity and Stability of K0,4.3.2 Direct limits of *-algebras , 4.3.3 C*--algebraic inductive limits, 5 K1-Functor and the Index Map 51, 5.2 The Index Map p.55 5.2.1 Fredholm index 6 Bott Periodicity and the Exact Sequence of K-Theory 67 6.1 Higher K-Groups 6.1.1 The suspension functor, The long exact sequence of K-theory, 6.2 Bott Periodicity Definition of the Bott map, p.69 6.2.2 The periodicity theorem 6.3 The 6-Term Exact Sequence p.74 6.3.1 The 6-term exact sequence of K-theory 6.3.2 The exponential map p. 75 6.3.3 An explicit form of the exponential map 7 Tools for the computation of K-groups 83 7.1 Crossed products, the Thom-Connes isomorphism and the Pimsner-Voiculescu sequence 7.1.1 Crossed products p.83 7.1.2 Crossed products by R and by Z, p. 84 7.1.3 Irrational rotation algebras 7.2 The Mayer-Vietoris sequence, p.86 7.3 The Kunneth formula

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