Title: A hierarchy of Banach spaces with C(K) Calkin algebras

Authors: Pavlos Motakis, Daniele Puglisi and Despoina Zisimopoulou

Issue: Volume 65 (2016), Issue 1, 39-67


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 $C(K)$.