Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

Let ω1 be the first 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 final column is continuous when viewed as a scalar-valued function on [0,ω1] defines 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 finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).