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Let $\omega_1$ be the first uncountable ordinal. By a result of Rudin,bounded operators on the Banach space $C([0,\omega_1])$ have a naturalrepresentation as $[0,\omega_1]\times 0,\omega_1]$-matrices. Loy and Willisobserved that the set of operators whose final column is continuous when viewedas a scalar-valued function on $[0,\omega_1]$ defines a maximal ideal ofcodimension one in the Banach algebra $\mathscr{B}(C([0,\omega_1]))$ of boundedoperators on $C([0,\omega_1])$. We give a coordinate-free characterization ofthis ideal and deduce from it that $\mathscr{B}(C([0,\omega_1]))$ contains noother maximal ideals. We then obtain a list of equivalent conditions describingthe strictly smaller ideal of operators with separable range, and finally weinvestigate the structure of the lattice of all closed ideals of$\mathscr{B}(C([0,\omega_1]))$.

Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,\omega_1])$