For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^*$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.

中文翻译：

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与拓扑绝缘体相关的 Hilbert 模块中的 Good Wannier 基

对于具有离散群作用的流形上的一大类物理相关算子，我们证明了其谱子空间的光滑良好局部化万尼尔函数的基的（不）存在的一般结果。事实证明，这等效于某个 Hilbert 模块在与谱子空间规范相关的 $C^*$-代数群上的自由度。这使 $K$ 理论方法发挥作用，并证明了它们作为物理学中拓扑绝缘体不变量的重要性。