Tomasz Kania (Lancaster) - "Closed ideals of the Banach algebra of bounded operators on the Banach space C [0, ω_1]"

Abstract

Let ω₁ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C [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 of bounded operators on C [0, ω₁]. We give a coordinate-free characterisation of this ideal and deduce that it is the 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. This is joint work with Niels Jakob Laustsen.