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Von Neumann spectra near zero. (English) Zbl 0751.58039
Let \(M\) be a Riemannian \(\Gamma\)-manifold with a discrete infinite group \(\Gamma\) of isometries of \(M\). Let us suppose that \(\Gamma\) acts freely on \(M\) and that \(M/\Gamma\) is a compact manifold. If \(L^ 2\wedge^ k(M)\) is the Hilbert space of all square integrable exterior \(k\)-forms on \(M\), then it is well known that the Laplacian \(\Delta_ k\) is essentially self-adjoint in \(L^ 2\wedge^ k(M)\), i.e. its closure \(\bar\Delta_ k\) is a self-adjoint operator in \(L^ 2\wedge^ k(M)\), so we have the spectral decomposition: \(\bar\Delta_ k=\int\lambda dE_ \lambda^{(k)}\) and therefore we can define the spectrum distribution function \(N_ k(\lambda)\) by: \(N_ k(\lambda)=\text{Tr}_{\Gamma}(E_ \lambda^{(k)})\). The most important characteristic of \(N_ k(\lambda)\) is due to M. Atiyah, Astérisque 32-33, 43-72 (1976; Zbl 0323.58015), namely that: \(\bar b_ k=\lim_{\lambda\to 0_ +}N_ k(\lambda)\), where \(\bar b_ k\) is the \(L^ 2\)-Betti number of \(M\).
In the paper under review the authors prove that the asymptotics of \(N_ k(\lambda)\) at \(\lambda=0\) is in fact a homotopy invariant of \(M\). This is done by expressing this asymptotics as a chain homotopy invariant of the De Rham \(L^ 2\)-complex of \(M\) in the category of Hilbert spaces and bounded linear operators.

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
Zbl 0323.58015
Full Text: DOI EuDML
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