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This monograph develops the theory of Besov spaces for abelian group actionson semifinite von Neumann algebras and then proves Peller criteria fortraceclass properties of associated Hankel operators. This allows to extendknown index theorems to symbols lying in Sobolev or Besov spaces. The dualitytheory for pairings over the smooth Toeplitz extension is developed in detail.Numerous applications to solid state systems are presented. In particular, abulk-boundary correspondence is obtained for insulators with edges ofirrational angles and for chiral semimetals having a pseudogaps. The latterimplies the existence of flat bands of edge for tight-binding graphene modelsand shows how the density of surface states is expressed in terms of weak Chernnumbers of the system without boundaries.

Harmonic analysis in operator algebras and its applications to index theory and topological solid state systems