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

Abstract:

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.