  Let $S$ be the semigroup with identity, generated by $x$ and $y$, subject to
$y$ being invertible and $yx=xy^2$. We study two Banach algebra completions of
the semigroup algebra $\mathbb{C}S$. Both completions are shown to be
left-primitive and have separating families of irreducible infinite-dimensional
right modules. As an appendix, we offer an alternative proof that $\mathbb{C}S$
is left-primitive but not right-primitive. We show further that, in contrast to
the completions, every irreducible right module for $\mathbb{C}S$ is finite
dimensional and hence that $\mathbb{C}S$ has a separating family of such
modules.
