  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 natural
representation as $[0,\omega_1]\times 0,\omega_1]$-matrices. Loy and Willis
observed that the set of operators whose final column is continuous when viewed
as a scalar-valued function on $[0,\omega_1]$ defines a maximal ideal of
codimension one in the Banach algebra $\mathscr{B}(C([0,\omega_1]))$ of bounded
operators on $C([0,\omega_1])$. We give a coordinate-free characterization of
this ideal and deduce from it that $\mathscr{B}(C([0,\omega_1]))$ 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
$\mathscr{B}(C([0,\omega_1]))$.
