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

For every well-founded tree T having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a L∞-space XT . We prove that, for each such tree T , the Calkin algebra of XT is homomorphic to C(T ), the algebra of continuous functions defined on T , equipped with the usual topology. We use this fact to conclude that, for every countable compact metric space K, there exists a L∞-space whose Calkin algebra is isomorphic, as a Banach algebra, to C(K).