Kania, Tomasz and Koszmider, Piotr and Laustsen, Niels
(2014)
*A weak*-topological dichotomy with applications in operator theory.*
Transactions of the London Mathematical Society, 1 (1).
pp. 1-28.
ISSN 2052-4986

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## 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 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.