Morita equivalence of two $\ell^p$ Roe-type algebras

Abstract

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.