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

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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: Journal 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: /dk/atira/pure/researchoutput/libraryofcongress/qa
Subjects:
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 49558
Deposited By: ep_importer_pure
Deposited On: 07 Sep 2011 08:29
Refereed?: Yes
Published?: Published
Last Modified: 27 Feb 2020 01:46
URI: https://eprints.lancs.ac.uk/id/eprint/49558

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