This is an announcement for the paper "Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,\omega_1])$" by Tomasz Kania and Niels Jakob Laustsen.

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

Archive classification: math.FA

Mathematics Subject Classification: Primary 47L10, 46H10, Secondary 47L20, 46B26, 47B38

Submitted from: t.kania@lancaster.ac.uk

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1112.4800

or

http://arXiv.org/abs/1112.4800