TY - JOUR

T1 - Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals

AU - Kania, Tomasz

AU - Laustsen, Niels

N1 - First published in Proceedings of the American Mathematical Society in Vol. 143, (2015), published by the American Mathematical Society

PY - 2015

Y1 - 2015

N2 - 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, 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

AB - 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, 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

U2 - 10.1090/S0002-9939-2015-12480-X

DO - 10.1090/S0002-9939-2015-12480-X

M3 - Journal article

VL - 143

SP - 2585

EP - 2596

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

ER -