A hierarchy of Banach spaces with C(K) Calkin algebras Pavlos MotakisDaniele PuglisiDespoina Zisimopoulou 46B0346B2546B28Calkin AlgebrasBourgain-Delbaen method$\\mathcal{L}_\\infty$ spaces For every well-founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct an $\mathcal{L}_{\infty}$-space $X_{\mathcal{T}}$. We prove that, for each such tree $\mathcal{T}$, the Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$, equipped with the usual topology. We use this fact to conclude that, for every countable compact metric space $K$, there exists a $\mathcal{L}_{\infty}$-space whose Calkin algebra is isomorphic, as a Banach algebra, to $C(K)$. Indiana University Mathematics Journal 2016 text pdf 10.1512/iumj.2016.65.5756 10.1512/iumj.2016.65.5756 en Indiana Univ. Math. J. 65 (2016) 39 - 67 state-of-the-art mathematics http://iumj.org/access/