  In this paper we study some new von Neumann spectral invariants associated to
the Laplacian acting on L^2 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 L^2 theta and
L^2 zeta functions defined by metric dependent combinatorial Laplacians acting
on $L^2$ cochains associated with a triangulation of the manifold, converge
uniformly to their analytic counterparts, as the mesh of the triangulation goes
to zero.
