2023/03/31 by Kordyukov, Yu., Manuilov, V.
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2304.00125
Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset D in a complete Riemannian manifold X. They show that, under certain geometric conditions on X, the class of the orthogonal projection onto the span of such a Wannier basis in the K-theory of the Roe algebra C^*(X) is trivial. In this paper, we clarify the geometric conditions on X, which guarantee triviality of the K-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of D in the K-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the K-theory class of any Wannier projection on a connected proper measure space X of bounded geometry with a uniformly discrete set of localization centers.