A weak*-topological dichotomy with applications in operator theory

TL;DR

This paper establishes a dichotomy for weak*-compact sets in the dual of a specific Banach space related to countable ordinals, leading to new insights into the structure of operators and ideals in that space.

Contribution

It proves a weak*-topological dichotomy for subsets of the dual space of C_0[0,ω_1), characterizes quotients and ideals, and resolves an open problem about the Loy-Willis ideal.

Findings

Weak*-compact sets are either uniformly Eberlein compact or contain a homeomorphic copy of [0,ω_1].

A quotient of C_0[0,ω_1) embeds in a Hilbert-generated Banach space or decomposes accordingly.

The Loy-Willis ideal has a bounded left approximate identity, resolving an open problem.

Abstract

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. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,\omega_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,\omega_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,\omega_1)$ can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of $C_0[0,\omega_1)$ and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy-Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on $C_0[0,\omega_1)$. As a consequence, we find that this ideal has a bounded left approximate identity, thus resolving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space $C_0[0,\omega_1)$ and the operators acting on it.