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The goal of this paper is to study band-dominated operators on Banach spaceswith Schauder basis with respect to uniformly locally finite metric spaces aswell as the Banach algebras generated by them: the so called uniform Roealgebras. We investigate several kinds of isomorphisms between those Banachspaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc)and prove several rigidity results on when a certain kind of isomorphismbetween the uniform Roe algebras implies that the base metric spaces are(bijectively) coarsely equivalent.

On Banach algebras of band-dominated operators and their order structure