Let S be the semigroup with identity, generated by x and y, subject to
y being invertible and yx=xy2. We study two Banach algebra completions of
the semigroup algebra CS. 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 CS
is left-primitive but not right-primitive. We show further that, in contrast to
the completions, every irreducible right module for CS is finite
dimensional and hence that CS has a separating family of such
modules.Comment: 14 pages. To appear, with minor changes, in the Proceedings of the
Edinburgh Mathematical Societ