K-Theory for C*-Algebras

Due to Swan’s Theorem 5.2.12 , topological K-theory for compact Hausdorff spaces X may be equivalently defined in terms of isomorphism classes of finitely generated projective (f.g.p.) modules over \(C(X)\) , and this suggests a definition of \(\mathrm {K}_0\) -theory of C*-algebras or indeed of any ring. At the latter level of generality the resulting theory is called algebraic K-theory, while when specialized to C*-algebras it is often called operator K-theory. Operator K-theory retains the Bott Periodicity phenomenon of topological K-theory, while algebraic K-theory does not, so the two differ in their treatment of higher K-groups. Operator K-theory is Morita invariant, and is the correct homology theory for studying the “noncommutative spaces” of Noncommutative Geometry.