  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.
