Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces

For a scattered, locally compact Hausdorff space $K$, we prove that the essential norm on the Calkin algebra \break $\mathcal{B}(C_0(K))\mathrm{/}\mathcal{K}(C_0(K))$ is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on $c_0$ through non-compact operators $T:C_0(K)\to X$, where $X$ is any Banach space that does not contain a copy of ${\mathrm{\ell }}_1$ or whose dual unit ball is weak${}^{\ast }$ sequentially compact. It follows that, for every ordinal $\alpha$, the algebras $\mathcal{B}(C[0,\alpha ]))$ and $\mathcal{B}(C[0,\alpha ]))\mathrm{/}\mathcal{K}(C[0,\alpha ]))$ have an unique algebra norm.