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

TL;DR

This paper investigates the structure of maximal left ideals in the algebra of bounded operators on infinite-dimensional Banach spaces, providing conditions under which certain ideals are finitely generated or not.

Contribution

It answers two open questions about maximal left ideals, showing that most infinite-dimensional Banach spaces have non-finitely generated maximal left ideals and characterizing when finitely-generated ideals are of a specific form.

Findings

Most infinite-dimensional Banach spaces have a maximal left ideal that is not finitely generated.

Finitely-generated, maximal left ideals are of a specific form if and only if they do not contain the finite-rank operators.

The positive answer to the second question holds for many but not all Banach spaces.

Abstract

We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ 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 (Q1) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (Q2) is positive for many, but not all, Banach spaces.