Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals
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- by Tomasz Kania and Niels Jakob Laustsen PDF
- Proc. Amer. Math. Soc. 143 (2015), 2585-2596 Request permission
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, 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.References
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Additional Information
- Tomasz Kania
- Affiliation: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom — and — Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
- MR Author ID: 976766
- ORCID: 0000-0002-2002-7230
- Email: tomasz.marcin.kania@gmail.com
- Niels Jakob Laustsen
- Affiliation: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom
- MR Author ID: 640805
- Email: n.laustsen@lancaster.ac.uk
- Received by editor(s): April 17, 2013
- Received by editor(s) in revised form: February 4, 2014
- Published electronically: February 5, 2015
- Communicated by: Thomas Schlumprecht
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2585-2596
- MSC (2010): Primary 46H10, 47B38, 47L10; Secondary 06F30, 46B26, 47L20
- DOI: https://doi.org/10.1090/S0002-9939-2015-12480-X
- MathSciNet review: 3326039