  For differential operators which are invariant under the action of an abelian
group Bloch theory is the preferred tool to analyze spectral properties. By
shedding some new non-commutative light on this we motivate the introduction of
a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It
relates properties of C*-algebras to spectral properties of module operators
such as band structure, weak genericity of cantor spectra, and absence of
discrete spectrum. It applies e.g. to differential operators invariant under a
projective group action, such as Schroedinger, Dirac and Pauli operators with
periodic magnetic field, as well as to discrete models, such as the almost
Matthieu equation and the quantum pendulum.
