Abstract. We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada [Sun]. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in [MY]. The result is generalized to other self adjoint operators with finite propagation. 1.

Given a C ∗-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [16], and show that A R is a C ∗-algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with improper Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on A R, denoted by AR, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (non-atomic) trace τ have been defined as traces on M − M-bimodules of unbounded τ-measurable operators [5], type II1 singular traces for a C ∗-algebra A with a semicontinuous semifinite (non-atomic) trace τ are defined here as traces on A − A-bimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗-algebras is contained in [6].

A semicontinuous semifinite trace is constructed on the C*-algebra

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