Fix $\mathcal H$ to be the separable infinite-dimensional Hilbert space. A bounded operator on $\mathcal H$ is Fredholm if and only if it is a unit in $B(\mathcal H)/K(\mathcal H)$, the Calkin algebra obtained by modding out bounded operators by compact ones. Because the index is invariant under compact perturbations, the Fredholm index function is well-defined on $(B(\mathcal H)/K(\mathcal H))^\times$.
Is the Fredholm index attainable from the Calkin algebra? It should present as connected components, but is the numerical invariant present somehow? Is there a particular representation of the algebra that recovers it?