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$C^*$-algebras associated with textile dynamical systems

A $C^*$-symbolic dynamical system $({\cal A}, ρ, Σ)$ is a finite family $\{ρ_α\}_{α\inΣ}$ of endomorphisms of a $C^*$-algebra ${\cal A}$ with some conditions. It yields a $C^*$-algebra ${\cal O}_ρ$ from an associated Hilbert $C^*$-bimodule. In this paper, we will extend the notion of $C^*$-symbolic dynamical system to $C^*$-textile dynamical system $({\cal A}, ρ, η, {Σ^ρ}, {Σ^η}, κ)$ which consists of two $C^*$-symbolic dynamical systems $({\cal A}, ρ, {Σ^ρ})$ and $({\cal A}, η, {Σ^η})$ with certain commutation relations $κ$ between their endomorphisms $\{ρ_α\}_{α\in Σ^ρ}$ and $\{η_a \}_{a \in Σ^η}$. $C^*$-textile dynamical systems yield two-dimensional subshifts and $C^*$-algebras ${\cal O}^κ_{ρ,η}$. We will study the structure of the algebras ${\cal O}^κ_{ρ,η}$ and present its K-theory formulae.