Groupoids, von Neumann algebras and the integrated density of states

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrodinger operators on manifolds, quantum percolation and quasi-crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.