Von Neumann spectra near the spectral gap

Details Von Neumann spectra near the spectral gap Von Neumann spectra near the spectral gap

1996-10-29

arxiv.org/abs/dg-ga/9610019v1

Mathematics

Differential Geometry

LaTeX, 31 pages, to appear in Bull. de la Soc. Math. de France

Scientific paper

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.

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