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

For a scattered, locally compact Hausdorff space , we prove that the essential norm on the Calkin algebra is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on through noncompact operators , where is any Banach space that does not contain a copy of or whose dual unit ball is sequentially compact. It follows that, for every ordinal , the algebras and have a unique algebra norm.