A hierarchy of Banach spaces with $C(K)$ Calkin Algebras

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 a ${\mathcal{L}}_{\mathrm{\infty }}$-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 ${\mathcal{L}}_{\mathrm{\infty }}$-space whose Calkin algebra is isomorphic, as a Banach algebra, to $C(K)$.