1994/03/31 by Johannes Kellendonk · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebra over a field #Cartesian product #Combinatorics #Commutative property #Geometry #Invariant (physics) #Mathematics #Mathematics and Applications #Noncommutative geometry #Observable #Physics #Pure mathematics #Quantum mechanics #Quasicrystal Structures and Properties #Substitution tiling #Tessellation (computer graphics) #cond-mat.stat-mech #hep-th #math.OA
paper · pdf · doi:10.1142/s0129055x95000426
45 pages, 3 figures (few corrections in section 4)
arxiv created 1994/04/22 · openalex publication_date 1995/10/01 · arxiv updated 2016/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
To a given tiling a noncommutative 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 Schrödinger 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. The part of this image which is given by an invariant measure on the hull of the tiling 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.