[Paper Review] $C^*$-algebras and Fell bundles associated to a textile system

Valentin Deaconu|arXiv (Cornell University)|Dec 30, 2009

Advanced Operator Algebra Research7 references4 citations

TL;DR

This paper introduces $C^*$-algebras and Fell bundles associated with textile systems—combinatorial structures modeling two-dimensional shifts of finite type via Wang tilings. By constructing two families of graph $C^*$-algebras ($\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$) from one-dimensional shifts derived from the textile system, the authors compute K-theory invariants that encode dynamical complexity, and under path-lifting conditions, define groupoid morphisms and Fell bundles to capture the structure of the two-dimensional shift 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 $p,q:G o H$, 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 $π, ρ:Γ(G) o Γ(H)$ between the corresponding étale groupoids of $G$ and $H$. We define two families ${\mathcal A}(m,n)$ and $\bar{\mathcal A}(m,n)$ of $C^*$-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.

Motivation & Objective

• To establish a connection between two-dimensional shifts of finite type and operator algebras using textile systems.
• To define and compute $C^*$-algebra invariants ($\mathcal{A}(m,n)$, $\bar{\mathcal{A}}(m,n)$) derived from one-dimensional shifts associated with textile systems.
• To introduce Fell bundles that encode the complexity of two-dimensional shifts under additional structural conditions.
• To generalize known results from one-dimensional shifts and rank $d$ graphs to the two-dimensional case using combinatorial and algebraic tools.

Proposed method

• Construct two families of $C^*$-algebras, $\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$, as graph algebras from one-dimensional shifts derived from the textile system.
• Use the path-lifting property of morphisms $p,q: G \to H$ to induce groupoid morphisms $\pi, \rho: \Gamma(G) \to \Gamma(H)$ between the étale groupoids of the graphs.
• Define dual textile systems $\bar{T} = (\bar{G}, \bar{H}, s, r)$ by swapping the roles of $p,q$ and $s,r$ to generate alternative graph structures.
• Apply techniques from graph $C^*$-algebras and Fell bundle theory to analyze the algebraic structure of the two-dimensional shift.
• Utilize K-theory of the constructed $C^*$-algebras as invariants for the two-dimensional shift, particularly under transitive and non-permutation conditions.
• Analyze specific examples, including full shifts, Golden Mean shifts, and rank two graph shifts, to illustrate the construction and invariants.

Experimental results

Research questions

• RQ1 How can $C^*$-algebras be systematically associated with two-dimensional shifts of finite type via textile systems?
• RQ2 What is the role of the path-lifting property in inducing groupoid morphisms between the étale groupoids of the underlying graphs?
• RQ3 How do the K-theory groups of the constructed $C^*$-algebras $\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$ serve as invariants for the two-dimensional shift?
• RQ4 Under what conditions can Fell bundles be defined to encode the dynamical complexity of textile shifts?
• RQ5 How do the invariants derived from the $C^*$-algebras compare to classical invariants like entropy and zeta functions in known examples?

Key findings

• The $C^*$-algebras $\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$ are constructed as graph algebras from one-dimensional shifts derived from the textile system, providing a bridge between 2D shifts and operator algebras.
• When the morphisms $p$ and $q$ have the path-lifting property, groupoid morphisms $\pi, \rho: \Gamma(G) \to \Gamma(H)$ are induced between the étale groupoids of $G$ and $H$.
• The K-theory groups of $\mathcal{A}(m,n)$ and $\bar{\mathcal{A}}(m,n)$ serve as invariants for the two-dimensional shift, with explicit computations provided in specific cases such as the full shift and Golden Mean shift.
• Fell bundles are defined under additional hypotheses, encoding the complexity of the two-dimensional shift through algebraic structures generalizing groupoid $C^*$-algebras.
• The construction generalizes known results for rank $d$ graphs and provides a framework for studying higher-dimensional shifts using operator algebra techniques.
• Examples including the full shift, Golden Mean shift, and rank two graph shifts demonstrate the applicability and invariance properties of the constructed algebras.