  In this paper we consider bounded operators on infinite graphs, in particular
Cayley graphs of amenable groups. The operators satisfy an equivariance
condition which is formulated in terms of a colouring of the vertex set of the
underlying graph. In this setting it is natural to expect that the integrated
density of states (IDS), or spectral distribution function, exists. We show
that it can be defined as the uniform limit of approximants associated to
finite matrices. The proof is based on a Banach space valued ergodic theorem
which even allows explicit convergence estimates. Our result applies to a
variety of group structures and colouring types, in particular to periodic
operators and percolation-type Hamiltonians.
