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)

Request Permissions   Purchase Content 
 
 

 

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?)

  • [1] Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298 (2006h:46005)
  • [2] Dale E. Alspach, $ C(K)$ norming subsets of $ C[0,\,1]$, Studia Math. 70 (1981), no. 1, 27-61. MR 646959 (83h:46032)
  • [3] D. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), no. 1, 64-92. MR 0440349 (55 #13224)
  • [4] C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53-62. MR 0113132 (22 #3971)
  • [5] Alistair Bird, Graham Jameson, and Niels Jakob Laustsen, The Giesy-James theorem for general index $ p$, with an application to operator ideals on the $ p$th James space, J. Operator Theory 70 (2013), no. 1, 291-307. MR 3085829, https://doi.org/10.7900/jot.2011aug11.1936
  • [6] J. Bourgain, The Szlenk index and operators on $ C(K)$-spaces, Bull. Soc. Math. Belg. Sér. B 31 (1979), no. 1, 87-117. MR 592664 (83j:46027)
  • [7] Philip A. H. Brooker, Direct sums and the Szlenk index, J. Funct. Anal. 260 (2011), no. 8, 2222-2246. MR 2772370 (2012f:46024), https://doi.org/10.1016/j.jfa.2010.12.016
  • [8] Philip A. H. Brooker, Asplund operators and the Szlenk index, J. Operator Theory 68 (2012), no. 2, 405-442. MR 2995728
  • [9] John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • [10] D. Dosev and W. B. Johnson, Commutators on $ \ell _\infty $, Bull. Lond. Math. Soc. 42 (2010), no. 1, 155-169. MR 2586976 (2011d:47084), https://doi.org/10.1112/blms/bdp110
  • [11] Petr Hájek and Gilles Lancien, Various slicing indices on Banach spaces, Mediterr. J. Math. 4 (2007), no. 2, 179-190. MR 2340479 (2008f:46022), https://doi.org/10.1007/s00009-007-0111-4
  • [12] Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 26, Springer, New York, 2008. MR 2359536 (2008k:46002)
  • [13] Tomasz Kania, Piotr Koszmider, and Niels Jakob Laustsen, A weak$ {}^*$-topological dichotomy with applications in operator theory, Trans. London Math. Soc. 1 (2014), no. 1, 1-28. MR 3247078
  • [14] Tomasz Kania and Niels Jakob Laustsen, Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $ C([0,\omega _1])$, J. Funct. Anal. 262 (2012), no. 11, 4831-4850. MR 2913688, https://doi.org/10.1016/j.jfa.2012.03.011
  • [15] A. Pełczyński, On $ C(S)$-subspaces of separable Banach spaces, Studia Math. 31 (1968), 513-522. MR 0234261 (38 #2578)
  • [16] C. Samuel, Indice de Szlenk des $ C(K)$ ($ K$ espace topologique compact dénombrable), Seminar on the geometry of Banach spaces, Vol. I, II (Paris, 1983), Publ. Math. Univ. Paris VII, vol. 18, Univ. Paris VII, Paris, 1984, pp. 81-91 (French). MR 781569 (87b:46026)
  • [17] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53-61. MR 0227743 (37 #3327)

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