On the spectrum of periodic elliptic operators. (English) Zbl 0759.35016

Brüning, Jochen; Sunada, Toshikazu

Nagoya Math. J. 126, 159-171 (1992).

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

Keywords:

selfadjoint operators; symmetric elliptic operator; complete Riemannian manifold; discontinuous isometric action of a group; Kadison property; band structure

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