Delone dynamical systems and associated random operators

TL;DR

This paper studies the topological and ergodic properties of Delone dynamical systems, explores their associated groupoids and von Neumann algebras, and discusses a Shubin trace formula for certain operators.

Contribution

It provides a detailed analysis of the von Neumann algebras arising from Delone systems and introduces a trace formula for tight binding operators.

Findings

Characterization of von Neumann algebras from Delone systems

Representation of groupoids on direct integrals with non-constant fibers

Derivation of a Shubin trace formula for specific operators

Abstract

We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via non-commutative-integration theory these representations give rise to von Neumann algebras of random operators. Features of these algebras and operators are discussed. Restricting our attention to a certain subalgebra of tight binding operators, we then discuss a Shubin trace formula.