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

TL;DR

This paper proves Morita equivalence between two $^p$ Banach algebras associated with metric spaces of bounded geometry, showing they share $K$-theory, and investigates the properties of an $^p$ uniform coarse assembly map.

Contribution

It establishes Morita equivalence of $^p$ uniform Roe and algebra, and analyzes the surjectivity of the associated coarse assembly map.

Findings

The $^p$ uniform Roe algebra and the $^p$ uniform algebra are Morita equivalent.

Both algebras have the same $K$-theory.

The $^p$ uniform coarse assembly map is not always surjective.

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.