Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a non-metrisable compact, Hausdorff space, then the unit sphere of $X=C(K)$ also contains such a subset; if moreover $K$ is perfectly normal, then one can find such a set with cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat

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 formfor 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.