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On the spectrum of periodic elliptic operators. (English) Zbl 0759.35016
We consider a symmetric elliptic operator $$D$$ on a complete Riemannian manifold which admits a properly discontinuous isometric action of a group $$\Gamma$$, with compact quotient. We assume that $$D$$ commutes with the action of $$\Gamma$$. We associate with this situation the $$C^*$$- algebra $$C_ r^*(\Gamma)\oplus{\mathcal K}$$, where $$C_ r^*(\Gamma)$$ is the reduced $$C^*$$-algebra of $$\Gamma$$ and $${\mathcal K}$$ the compact operators on a suitable Hilbert space. This $$C^*$$-algebra has a natural trace; we say that it has the Kadison property if the traces of all nontrivial orthogonal projections have a positive lower bound. We prove that the Kadison property implies band structure for the spectrum of (the unique self-adjoint extension of) $$D$$ in the sense that the spectrum is a locally finite union of mutually disjoint, closed (possibly degenerate) intervals in $$\mathbb{R}$$.
Reviewer: J.Brüning

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P05 General topics in linear spectral theory for PDEs 47C15 Linear operators in $$C^*$$- or von Neumann algebras
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