A weak*-topological dichotomy with applications in operator theory (Q2921572)

Denote by \([0, \omega_{1})\) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let \(C_{0}[0,\omega_{1})\) be the Banach space of scalar-valued, continuous functions which are defined on \([0, \omega_{1})\) and vanish eventually. The authors show that a \(\text{weak}^{*}\)-compact subset of the dual space of \(C_{0}[0,\omega_{1})\) is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval \([0, \omega_{1})\). This dichotomy yields a unifying approach to most of the existing studies of the Banach space \(C_{0}[0,\omega_{1})\) and the Banach algebra \(\mathcal{B}(C_{0}[0,\omega_{1}))\) of bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. In particular, the authors solve a problem left open by \textit{R. J. Loy} and \textit{G. A. Willis} [J. Lond. Math. Soc., II. Ser. 40, No. 2, 327--346 (1989; Zbl 0651.47035)].