Morita equivalence of two ᵖ Roe-type algebras

Given a metric space with bounded geometry, one may associate with it the ^p uniform Roe algebra and the ^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 p<. As a consequence, these two Banach algebras have the same K -theory. We then define an ^p uniform coarse assembly map taking values in the K -theory of the ^p uniform Roe algebra and show that it is not always surjective.