Good Wannier bases in Hilbert modules associated to topological insulators

Abstract

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 well-localized 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 with the spectral subspace. This brings into play K-theoretic methods and justifies their importance as invariants of topological insulators in physics.