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 weakly* compact subset of the dual space of C₀[0,ω₁) is either uniformly Eberlein compact, or it contains a homeomorphic copy 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 operators acting on it, and it leads to several new results, as well as 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. Moreover, we obtain a list of eight equivalent conditions describing the Loy-Willis ideal M, which is the unique maximal ideal of B(C₀[0,ω₁)). Among the consequences of the latter result is that M has a bounded left approximate identity, thus resolving a problem left open by Loy and Willis.