Dales, H.G. and Kania, Tomasz and Kochanek, Tomasz and Koszmider, Piotr and Laustsen, Niels
(2013)
*Maximal left ideals of the Banach algebra of bounded operators on a Banach space.*
Studia Mathematica, 218 (3).
pp. 245-286.
ISSN 0039-3223

## Abstract

We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.