Random Operators and Crossed Products

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.