This monograph develops the theory of Besov spaces for abelian group actions
    on semifinite von Neumann algebras and then proves Peller criteria for
    traceclass properties of associated Hankel operators. This allows to extend
    known index theorems to symbols lying in Sobolev or Besov spaces. The duality
    theory for pairings over the smooth Toeplitz extension is developed in detail.
    Numerous applications to solid state systems are presented. In particular, a
    bulk-boundary correspondence is obtained for insulators with edges of
    irrational angles and for chiral semimetals having a pseudogaps. The latter
    implies the existence of flat bands of edge for tight-binding graphene models
    and shows how the density of surface states is expressed in terms of weak Chern
    numbers of the system without boundaries.

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