A weak∗‐topological dichotomy with applications in operator theory

Abstract

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] .