Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,{\omega }_1])$

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 $\mathcal{B}(C([0,{\omega }_1]))$ of bounded operators on $C([0,{\omega }_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathcal{B}(C([0,{\omega }_1]))$ 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 $\mathcal{B}(C([0,{\omega }_1]))$.

References in corpus (2)

• Commutators on ${\ell }_1$
• Dichotomy theorems for random matrices and closed ideals of operators on $({⨁}_{n=1}^{\infty }{\ell }_1^n{)}_{{\mathrm{c}}_0}$

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• Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals
• Banach spaces whose algebra of bounded operators has the integers as their $K_0$-group
• On the continuity of the composition operation on spaces of holomorphic mappings