# Closed operator ideals on the Banach space of continuous functions on the first uncountable ordinal

### Niels Lausten (U. Lancaster)

Tuesday 27th November, 2012 16:00-17:00 Maths 515

#### 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
co\-dimen\-sion one in the Banach algebra
$\mathscr{B}(C[0,\omega_1])$ of bounded operators on
$C[0,\omega_1]$. Our main result gives a list of co-ordinate-free
characterizations of this ideal, and implies in particular that
$\mathscr{B}(C[0,\omega_1])$ contains no other maximal ideals. We
also show that this maximal ideal has a bounded left approximate
identity, a result which complements Loy and Willis' construction of
a bounded right approximate identity.

This is joint work with Tomasz Kania (Lancaster) and Piotr Koszmider
(Warsaw).