Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals

Abstract

Denote by $[0,{\omega }_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,\alpha ]$ for $\alpha$ countable, and consider the Banach spaces $C_0[0,{\omega }_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,{\omega }_1)$ and $L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through $T$, if and only if the Szlenk index of $T$ is uncountable. This implies that the set ${\mathcal{S}}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $\mathcal{B}(C_0(L_0))$ of all bounded operators on $C_0(L_0)$, and that ${\mathcal{S}}_{C_0(L_0)}(C_0[0,{\omega }_1))$ is the second-largest proper ideal of $\mathcal{B}(C_0[0,{\omega }_1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous.

to appear in Proceedings of the American Mathematical Society

References in corpus (4)

• Commutators on ${\ell }_1$
• Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,{\omega }_1])$
• Direct sums and the Szlenk index
• A weak*-topological dichotomy with applications in operator theory

Cited by in corpus (3)

• Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
• When are full representations of algebras of operators on Banach spaces automatically faithful?
• Banach spaces whose algebra of bounded operators has the integers as their $K_0$-group