K0 of a C*-algebra

Summary

$K_0(A)$ is an abelian group attached to a C*-algebra $A$. It is built from projections in matrix algebras over $A$, counted up to Murray–von Neumann equivalence. The Grothendieck group construction then makes them into a group. For $A = C(X)$ with $X$ compact Hausdorff, it is the topological K-group $K^0(X)$ of vector bundles on $X$. It is the first invariant of operator K-theory and a key tool in the classification of C*-algebras.