  An odd index theorem for higher odd Chern characters of crossed product
algebras is proved. It generalizes the Noether-Gohberg-Krein index theorem.
Furthermore, a local formula for the associated cyclic cocycle is provided.
When applied to the non-commutative Brillouin zone, this allows to define
topological invariants for condensed matter phases from the chiral unitary (or
AIII-symmetry) class in the presence of strong disorder and magnetic fields
whenever the Fermi level lies in region of Anderson localization.
