  We investigate Banach algebras of convolution operators on the $L^p$ spaces
of a locally compact group, and their K-theory. We show that for a discrete
group, the corresponding K-theory groups depend continuously on $p$ in an
inductive sense. Via a Banach version of property RD, we show that for a large
class of groups, the K-theory groups of the Banach algebras are independent of
$p$.
