Uniqueness of the Maximal Ideal of the Banach Algebra of Bounded Operators on C([0, Ω1])

Let ω1 be the rst uncountable ordinal. A result of Rudin implies that bounded operators on the Banach space C([0, ω1]) of continuous functions on the ordinal interval [0, ω1] have a natural representation as [0, ω1]× [0, ω1]-matrices. Loy and Willis observed that the set of operators whose nal column is continuous when viewed as a scalar-valued function on [0, ω1] de nes 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 nally we investigate the structure of the lattice of all closed ideals of B(C([0, ω1])). Publisher's notice: this is the authors' version of a work that has been published in Journal of Functional Analysis 262 (2012), 4831 4850; DOI: 10.1016/j.jfa.2012.03.011. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication.