Lecture: Lecture Notes on the K-Theory of Operator Algebras
Authors:
Rainer Matthes Wojciech Szymanski,based primarily on: M. Roterdam, F. Larsen and N. J. Laustsen: An Introduction to K-Theory for C*-Algebras, and secondarily on:B. Blackadar: K-Theory for Operator Algebras,N. E. Wegge-Olsen: K-Theory and C*-Algebras
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
