Closed ideals of the Banach algebra of bounded operators on the Banach space C[0,\omega_1]

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 of bounded operators on C[0,\omega_1]. We give a coordinate-free characterisation of this ideal and deduce from it is a unique maximal ideal. 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 this algebra.