  Given a metric space with bounded geometry, one may associate with it the
$\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, both containing
information about the large scale geometry of the metric space. We show that
these two Banach algebras are Morita equivalent in the sense of Lafforgue for
$1\leq p<\infty$. As a consequence, these two Banach algebras have the same
$K$-theory. We then define an $\ell^p$ uniform coarse assembly map taking
values in the $K$-theory of the $\ell^p$ uniform Roe algebra and show that it
is not always surjective.
