2009/12/30 by Valentin Deaconu, Deaconu, Valentin
Mathematics · #46L05 #46L55 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L05 #msc:46L55
paper · pdf · doi:10.48550/arxiv.1001.0037
14 pages, 4 figures
arxiv created 2009/12/30 · arxiv updated 2010/01/14
The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms and automorphisms of topological Markov shifts. A textile system is given by two finite directed graphs G and H and two morphisms p,q:G→ H, with some extra properties. It turns out that a textile system determines a first quadrant two-dimensional shift of finite type, via a collection of Wang tiles, and conversely, any such shift is conjugate to a textile shift. In the case the morphisms p and q have the path lifting property, we prove that they induce groupoid morphisms π, ρ:Γ(G)→ Γ(H) between the corresponding étale groupoids of G and H. We define two families \mathcal A(m,n) and \mathcal A(m,n) of C^*-algebras associated to a textile shift, and compute them in specific cases. These are graph algebras, associated to some one-dimensional shifts of finite type constructed from the textile shift. Under extra hypotheses, we also define two families of Fell bundles which encode the complexity of these two-dimensional shifts. We consider several classes of examples of textile shifts, including the full shift, the Golden Mean shift and shifts associated to rank two graphs.