Factorizations of invertible operators and $K$-theory of $C^*$-algebras

Let $\mathcal{A}$ be a unital ${\text{C}}^{\ast}$-algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert ${\text{C}}^{\ast}$-module $\mathcal{H}_\mathcal{A}$, in particular on $\mathcal{H}={l^2}$, with the Fredholm index as an invariant. We then outline the isomorphisms ${K_0}(\mathcal{A}) \cong {\pi _{2k}}({[p]_0}) \cong {\pi _{2k}}({GL}_r^p(\mathcal{A}))$ and ${{K}_{1}}(\mathcal{A})\cong {{\pi }_{2k+1}}({[p]_0})\cong {{\pi }_{2k+1}}({GL}_r^p(\mathcal{A}))$ for $k \geq 0$, where ${[p]_0}$ denotes the class of all compact perturbations of a projection p in the infinite Grassmann space ${Gr}^{\infty}(\mathcal{A})$ and ${GL}_r^p(\mathcal{A})$ stands for the group of all those invertible operators on ${\mathcal{H}_\mathcal{A}}$ essentially commuting with p.