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

摘要

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 $[0,lpha]$ for $lpha$ 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, linear operator $T$ between any pair of these two Banach spaces fixes an isomorphic 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 $mathscr{S}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $mathscr{B}(C_0(L_0))$ of all bounded, linear operators on $C_0(L_0)$, and that $mathscr {S}_{C_0(L_0)}(C_0[0,omega_1))$ is the second-largest proper ideal of $mathscr{B}(C_0[0,omega _1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous. - See more at: http://www.ams.org/journals/proc/0000-000-00/S0002-9939-2015-12480-X/home.html#sthash.nZwAr45z.dpuf