$C^{\ast }$-algebras and Fell bundles associated to a textile system

Abstract

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 $p,q:G\to H$, 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 $\pi ,\rho :\Gamma (G)\to \Gamma (H)$ between the corresponding étale groupoids of $G$ and $H$. We define two families $\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$ of $C^{\ast }$-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.

14 pages, 4 figures

References in corpus (2)

• Crossed products of k-graph C*-algebras by Z^l
• Periodic 2-graphs arising from subshifts

Cited by in corpus (4)

• $C^{\ast }$-algebras associated with textile dynamical systems
• Cuntz-Krieger algebras associated with Hilbert $C^{\ast }$-quad modules of commuting matrices
• C*-algebras associated with Hilbert C*-quad modules of finite type
• C*-algebras associated with Hilbert C*-quad modules of C*-textile dynamical systems