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

Acuaviva Huertos, Antonio (2025) Factorizations and minimality of the Calkin algebra norm for C(K)-spaces. Journal of the London Mathematical Society. ISSN 0024-6107 (In Press)

Text (v3Text_File_Article_for_JLMS___Antonio_Acuaviva)
v3Text_File_Article_for_JLMS_Antonio_Acuaviva_3_.pdf - Accepted Version
Available under License Creative Commons Attribution.
Download (493kB)

Abstract

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.