C∗-algebras associated with textile dynamical systems

A C∗-symbolic dynamical system (A, ρ,Σ) is a finite family {ρα}α∈Σ of endomorphisms of a C∗-algebra A with some conditions. It yields a C∗-algebra Oρ from an associated Hilbert C∗-bimodule. In this paper, we will extend the notion of C∗-symbolic dynamical system to C∗-textile dynamical system (A, ρ, η,Σ,Σ, κ) which consists of two C∗-symbolic dynamical systems (A, ρ,Σ) and (A, η,Σ) with certain commutation relations κ between their endomorphisms {ρα}α∈Σρ and {ηa}a∈Ση . C∗-textile dynamical systems yield two-dimensional subshifts and C∗-algebras O ρ,η. We will study their structure of the algebras O ρ,η and present its K-theory formulae.