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Von Neumann spectra near the spectral gap

Identifieur interne : 002751 ( Istex/Curation ); précédent : 002750; suivant : 002752

Von Neumann spectra near the spectral gap

Auteurs : Alan L. Carey [Australie] ; Thierry Coulhon [France] ; Varghese Mathai [Australie] ; John Phillips [Canada]

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RBID : ISTEX:D3A19ABC400F673991C245370E203D5A751A342B

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Abstract

Abstract: 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.

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DOI: 10.1016/S0007-4497(98)80088-2

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ISTEX:D3A19ABC400F673991C245370E203D5A751A342B

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<div type="abstract" xml:lang="en">Abstract: 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.</div>
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