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

TL;DR

This paper characterizes the unique maximal ideal of the Banach algebra of bounded operators on $C([0,])$, showing it is the only maximal ideal and exploring the lattice of all closed ideals.

Contribution

It provides a coordinate-free characterization of the maximal ideal and describes the structure of all closed ideals in the algebra.

Findings

The maximal ideal is uniquely characterized and contains operators with continuous final columns.

Operators with separable range form a strictly smaller ideal with multiple equivalent conditions.

The structure of the lattice of all closed ideals is analyzed and described.

Abstract

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 $\mathscr{B}(C([0,\omega_1]))$.