Von Neumann spectra near the spectral gap

In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L² differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L² theta and L² zera functions defined by metric dependent combinatorial Laplacians acting on L₂ cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.