A weak∗‐topological dichotomy with applications in operator theory

Denote by the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let be the Banach space of scalar‐valued, continuous functions which are defined on and vanish eventually. We show that a weak‐compact subset of the dual space of is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval . This dichotomy yields a unifying approach to most of the existing studies of the Banach space and the Banach algebra 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 can either be embedded in a Hilbert‐generated Banach space, or it is isomorphic to the direct sum of and a subspace of a Hilbert‐generated Banach space; and we obtain several equivalent conditions describing the Loy–Willis ideal , which is the unique maximal ideal of , including the following: an operator belongs to if and only if it factors through the Banach space . Among the consequences of these characterizations of is that has a bounded left approximate identity; this resolves a problem left open by Loy and Willis.