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 $\mathscr{B}(C_0(K))/\mathscr{K}(C_0(K))$ 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 $\ell_1$ or whose dual unit ball is weak$^*$ sequentially compact. It follows that, for every ordinal $\alpha$, the algebras $\mathscr{B}(C[0,\alpha]))$ and $\mathscr{B}(C[0,\alpha]))/\mathscr{K}(C[0,\alpha]))$ have an unique algebra norm.