Maximal left ideals of the Banach algebra of bounded operators on a Banach space, in preparation

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 ∈ B(E) : Tx = 0} (∗) for some non-zero x ∈ E? Since the two-sided ideal F (E) of finite-rank operators is not contained in any of the maximal left ideals given by (∗), 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 F (E); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces