TY - JOUR

T1 - A weak*-topological dichotomy with applications in operator theory

AU - Kania, Tomasz

AU - Koszmider, Piotr

AU - Laustsen, Niels

N1 - © 2014 Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact the London Mathematical Society

PY - 2014/5/14

Y1 - 2014/5/14

N2 - 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.

AB - 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.

KW - Banach space

KW - continuous functions on the first uncountable ordinal interval

KW - scattered space

KW - uniform Eberlein compactness

KW - weak topology

KW - club set, stationary set

KW - Pressing Down Lemma, Delta-system Lemma

KW - Banach algebra of bounded operators

KW - maximal ideal

KW - bounded left approximate identity

U2 - 10.1112/tlms/tlu001

DO - 10.1112/tlms/tlu001

M3 - Journal article

VL - 1

SP - 1

EP - 28

JO - Transactions of the London Mathematical Society

JF - Transactions of the London Mathematical Society

SN - 2052-4986

IS - 1

ER -