Operator K-theory

Summary

Operator K-theory assigns to a C*-algebra $A$ two abelian groups $K_0(A)$ and $K_1(A)$. The group $K_0(A)$ is built from projections in matrix algebras over $A$, and $K_1(A)$ from unitaries up to homotopy. For the algebra $C(X)$ of continuous functions on a compact Hausdorff space $X$, it gives back the topological K-theory of $X$. It is a basic invariant in the classification of C*-algebras and in index theory.