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

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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).