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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.

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