Separable spaces of continuous functions as Calkin algebras

Abstract

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