  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.
  The general setting applies to many examples studied before while we lay
special emphasis on a new one: random Schr\"odinger operators on manifolds. In
this case, we show that the distribution function of the abstract density of
states coincides with the integrated density of states defined via an
exhaustion procedure.
  The presentation is as explicit as possible, yet abstract enough to cover
many examples studied so far. We extract parts of Connes' noncommutative
integration theory, present them in a self contained manner and use them in our
proofs. By reason of illustration, we parallely give more direct proofs for our
main example, whenever possible.
  (The second version contains a detailed discussion of the application of the
abstract results to Hamiltonians associated to tilings and percolation graphs.
