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Final published version, 357 KB, PDF document
Available under license: CC BY-NC
Final published version
Licence: CC BY-NC
Research output: Contribution to Journal/Magazine › Journal article › peer-review
<mark>Journal publication date</mark> | 14/05/2014 |
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<mark>Journal</mark> | Transactions of the London Mathematical Society |
Issue number | 1 |
Volume | 1 |
Number of pages | 28 |
Pages (from-to) | 1-28 |
Publication Status | Published |
<mark>Original language</mark> | English |
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.