  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.
