K-theory for C*-Algebras and for Topological Spaces

Abstract

K-theory is the study of a collection of abelian groups that are invariant to C*-algebras or to locally compact Hausdorff spaces. These groups are useful for distinguishing C*-algebras and topological spaces, and they are used in classification programs. In the thesis we will focus attention on the abelian groups $K_0(A)$ and $K^0(X)$ for a C*-algebra $A$ and for a locally compact Hausdorff space $X$. The group $K_0(C(X))$ is naturally isomorphic to $K^0(X)$ whenever $X$ is a locally compact Hausdorff space. The maps $K_0$ and $K^0$ are covariant and contravariant functors respectively, they satisfy some functorial properties that are useful for computation.