A weak∗-topological dichotomy with applications in operator theory

Denote by [0, ω1) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C0[0, ω1) be the Banach space of scalar-valued, continuous functions which are defined on [0, ω1) and vanish eventually. We show that a weak ∗compact subset of the dual space of C0[0, ω1) is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval [0, ω1]. This dichotomy yields a unifying approach to most of the existing studies of the Banach space C0[0, ω1) and the Banach algebra B(C0[0, ω1)) of bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. Specifically, we deduce that a Banach space which is a quotient of C0[0, ω1) can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C0[0, ω1) and a subspace of a Hilbert-generated Banach space; and we obtain several equivalent conditions describing the Loy–Willis ideal M , which is the unique maximal ideal of B(C0[0, ω1)), including the following: an operator belongs to M if and only if it factors through the Banach space ( ⊕ α<ω1 C[0, α])c0 . Among the consequences of these characterizations of M is that M has a bounded left approximate identity; this resolves a problem left open by Loy and Willis.