  We study automatic injectivity of surjective algebra homomorphisms from
$\mathscr{B}(X)$, the algebra of (bounded, linear) operators on $X$, to
$\mathscr{B}(Y)$, where $X$ is one of the following \emph{long} sequence
spaces: $c_0(\lambda)$, $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1
\leqslant p < \infty$) and $Y$ is arbitrary. \textit{En route} to the proof
that these spaces do indeed enjoy such a property, we classify two-sided ideals
of the algebra of operators of any of the aforementioned Banach spaces that are
closed with respect to the `sequential strong operator topology'.
