K1 of a C*-algebra

Summary

$K_1(A)$ is an abelian group attached to a C*-algebra $A$. It is built from unitary elements of matrix algebras over the unitization of $A$, counted up to homotopy. It measures how many essentially different ways a unitary can fail to be connected to the identity. For $A = C(X)$ with $X$ compact Hausdorff, it is the topological K-group $K^1(X)$.