A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

Abstract: 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.