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

TL;DR

This monograph develops harmonic analysis tools for operator algebras, extending index theorems and applying them to topological solid state systems, revealing new insights into edge states and bulk-boundary correspondence.

Contribution

It introduces Besov space theory for von Neumann algebras and extends index theorems to broader symbol classes, with significant applications in topological insulators and semimetals.

Findings

Bulk-boundary correspondence for insulators with irrational edges

Existence of flat edge bands in graphene models

Surface state density expressed via weak Chern numbers

Abstract

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.