Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$

Details Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$ Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$

2011-12-20

arxiv.org/abs/1112.4800v1

Mathematics

Functional Analysis

Scientific paper

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

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