Lancaster EPrints

Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

Kania, Tomasz and Laustsen, Niels (2012) Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1]). Journal of Functional Analysis, 262. pp. 4831-4850. ISSN 0022-1236

[img]
Preview
PDF - Submitted Version
Download (495Kb) | Preview

    Abstract

    Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).

    Item Type: Article
    Journal or Publication Title: Journal of Functional Analysis
    Additional Information: The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 262 (11), 2012, © ELSEVIER.
    Uncontrolled Keywords: Continuous functions on ordinals ; Bounded operators on Banach spaces ; Maximal ideal ; Loy–Willis ideal
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 49558
    Deposited By: ep_importer_pure
    Deposited On: 07 Sep 2011 09:29
    Refereed?: Yes
    Published?: Published
    Last Modified: 09 Oct 2013 13:14
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/49558

    Actions (login required)

    View Item