Non Commutative Geometry of Tilings and Gap Labelling

To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the tiling or its periodic identification. Its scaled ordered K_0-group furnishes the gap labelling of Schroedinger operators. The group is computed for one dimensional tilings and Cartesian products thereof. Its image under a state is investigated for tilings which are invariant under a substitution. Part of this image is given by an invariant measure on the hull of the tiling which is determined. The results from the Cartesian products of one dimensional tilings point out that the gap labelling by means of the values of the integrated density of states is already fully determined by this measure.