Non-commutative Bloch theory. An Overview

TL;DR

This paper introduces a non-commutative version of Bloch theory to analyze spectral properties of elliptic operators invariant under group actions, extending classical methods to more complex, non-commutative settings.

Contribution

It develops a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules, linking C*-algebra properties to spectral characteristics.

Findings

Relates C*-algebra properties to spectral features like band structure.

Shows applicability to operators with projective group symmetries.

Provides insights into spectra of magnetic Schrödinger operators.

Abstract

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies e.g. to differential operators invariant under a projective group action, such as Schroedinger operators with periodic magnetic field.