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We provide a construction of Roe (C*-)algebras of general coarse spaces interms of coarse geometric modules. This extends the classical theory of Roealgebras of metric spaces and gives a unified framework to deal with eitheruniform or non-uniform Roe algebras, algebras of operators of controlledpropagation, and algebras of quasi-local operators; both in the metric andgeneral coarse geometric setting. The key new definitions are that of coarsegeometric module and coarse support of operators between coarse geometricmodules. These let us construct natural bridges between coarse geometry andoperator algebras.  We then study the general structure of Roe-like algebras, and prove severalkey properties, such as admitting Cartan subalgebras or computing theirintersection with the compact operators. Lastly, we prove that assigning to acoarse space the K-theory groups of its Roe algebra(s) is a natural functorialoperation.

Roe algebras of coarse spaces via coarse geometric modules