Separable Spaces of Continuous Functions as Calkin Algebras

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 ${\mathcal{L}}_{\infty }$-space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded.

arXiv comment42 pages

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• Corona Rigidity
• On the complete separation of unique ${\ell }_1$ spreading models and the Lebesgue property of Banach spaces
• A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)