Roe algebras of coarse spaces via coarse geometric modules

Abstract

We provide a construction of Roe ($C^{\ast }$-)algebras of general coarse spaces in terms of coarse geometric modules. This extends the classical theory of Roe algebras of metric spaces and gives a unified framework to deal with either uniform or non-uniform Roe algebras, algebras of operators of controlled propagation and algebras of quasi-local operators, both in the metric and general coarse geometric settings. The key new definitions are those of coarse geometric module and coarse support of operators between coarse geometric modules. These let us construct natural bridges between coarse geometry and operator algebras. We then study the general structure of Roe-like algebras and investigate several structural properties, such as admitting (non-commutative) Cartan subalgebras or computing their intersection with the compact operators. Lastly, we prove that assigning to a coarse space the $K$-theory groups of its Roe algebra(s) is a natural functorial operation.