Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Tomasz Kania and Niels Jakob Laustsen
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
Published electronically: February 5, 2015
MathSciNet review: 3326039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46H10, 47B38, 47L10, 06F30, 46B26, 47L20

Retrieve articles in all journals with MSC (2010): 46H10, 47B38, 47L10, 06F30, 46B26, 47L20


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
Email: tomasz.marcin.kania@gmail.com

Niels Jakob Laustsen
Affiliation: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom
Email: n.laustsen@lancaster.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2015-12480-X
Keywords: Banach algebra, maximal ideal, bounded, linear operator, Szlenk index, continuous function, ordinal interval, order topology, Banach space, primary, complementably homogeneous
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
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society