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

Research output: Working paper

Published

Publication date2013
Number of pages9
Original languageEnglish

Abstract

Denote by [0,ω1) the set of countable ordinals, equipped with the order topology, let L0 be the disjoint union of the compact ordinal intervals [0,α] for α countable, and consider the Banach spaces C0[0,ω1) and C0(L0) consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces [0,ω1) and L0, 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 C0(L0) if and only if the identity operator on C0(L0) factors through T, if and only if the Szlenk index of T is uncountable. This implies that the set SC0(L0)(C0(L0)) of C0(L0)-strictly singular operators on C0(L0) is the unique maximal ideal of the Banach algebra B(C0(L0)) of all bounded operators on C0(L0), and that S0(0)CL(C0[0,ω1)) is the second-largest proper ideal of B(C0[0,ω1)). Moreover, it follows that the Banach space C0(L0) is primary and complementably homogeneous.