  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.
