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arXiv · 1112.4800

Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$

Tomasz Kania·Niels Jakob Laustsen
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Abstract

Let $ω_1$ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space $C([0,ω_1])$ have a natural representation as $[0,ω_1]\times 0,ω_1]$-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on $[0,ω_1]$ defines a maximal ideal of codimension one in the Banach algebra $\mathscr{B}(C([0,ω_1]))$ of bounded operators on $C([0,ω_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathscr{B}(C([0,ω_1]))$ 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 $\mathscr{B}(C([0,ω_1]))$.

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Tomasz Kania, Niels Jakob Laustsen. 2011-12-20. Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$. https://doi.org/10.1016/j.jfa.2012.03.011

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