Separable Spaces of Continuous Functions as Calkin Algebras

TL;DR

The paper demonstrates that for any compact metric space, there exists a Banach space whose Calkin algebra is isometrically isomorphic to the algebra of continuous functions on that space, using a modified Bourgain-Delbaen construction.

Contribution

It introduces a new method to realize any $C(K)$ as a Calkin algebra of a specially constructed Banach space, extending the understanding of Calkin algebras and Banach space theory.

Findings

Constructed Banach spaces with prescribed Calkin algebras

Extended Bourgain-Delbaen space techniques

Showed isometric isomorphism to $C(K)$ for any compact $K$

Abstract

It is proved that for every compact metric space $K$ there exists a Banach space $X$ whose Calkin algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is homomorphically isometric to $C(K)$. This is achieved by appropriately modifying the Bourgain-Delbaen $\mathscr{L}_\infty$-space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded.