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

TL;DR

This paper demonstrates that for certain scattered locally compact spaces, the essential norm on the Calkin algebra of $C_0(K)$-spaces is minimal, establishing uniqueness of algebra norms for these operator algebras.

Contribution

It proves the essential norm is minimal on the Calkin algebra for scattered spaces and establishes the uniqueness of algebra norms for these operator algebras.

Findings

Essential norm on the Calkin algebra is minimal for scattered spaces.

Unique algebra norm exists for $B(C[0,eta])$ and its quotient.

Quantitative factorization of the identity on $c_0$ through non-compact operators.

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.