Random Operators and Crossed Products

TL;DR

This paper explores the structure of crossed product von Neumann algebras, introduces a trace via noncommutative integration, and applies these concepts to random operators, including spectral measures and duality results.

Contribution

It provides new proofs for classical theorems, generalizes the integrated density of states, and establishes duality results for crossed products by Z.

Findings

The integrated density of states is a spectral measure in the periodic case.

New proofs for Bellissard and Testard's theorems on Plancherel-type results.

Duality results for crossed products by Z.

Abstract

This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes' non commutative integration theory and discuss Shubin's trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel Theorem. We show that the integrated density of states is a spectral measure in the periodic case, therby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z.