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Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformlylocalized Wannier basis with localization centers in an arbitrary uniformlydiscrete subset $D$ in a complete Riemannian manifold $X$. They show that,under certain geometric conditions on $X$, the class of the orthogonalprojection onto the span of such a Wannier basis in the $K$-theory of the Roealgebra $C^*(X)$ is trivial. In this short note, we clarify the geometricconditions on $X$, which guarantee triviality of the $K$-theory class of anyWannier projection. We show that this property is equivalent to triviality ofthe unit of the uniform Roe algebra of $D$ in the $K$-theory of its Roealgebra, and provide a geometric criterion for that. As a consequence, we provetriviality of the $K$-theory class of any Wannier projection on a connectedproper measure space $X$ of bounded geometry with a uniformly discrete set oflocalization centers, coarsely equivalent to $X$.

On topological obstructions to the existence of non-periodic Wannier bases