C⁎-algebras associated with Hilbert C⁎-quad modules of C⁎-textile dynamical systems

A C⁎C⁎-textile dynamical system (A,ρ,η,Σρ,Ση,κ)(A,ρ,η,Σρ,Ση,κ) consists of a unital C⁎C⁎-algebra AA, two families of endomorphisms {ρα}α∈Σρ{ρα}α∈Σρ and {ηa}a∈Ση{ηa}a∈Ση of AA and certain commutation relations κ   among them. It yields a two-dimensional subshift and a multistructure Hilbert C⁎C⁎-bimodule, which we call a Hilbert C⁎C⁎-quad module. We introduce a C⁎C⁎-algebra from the Hilbert C⁎C⁎-quad module as a two-dimensional analogue of Pimsner's construction of C⁎C⁎-algebras from Hilbert C⁎C⁎-bimodules. We study the C⁎C⁎-algebras defined by the Hilbert C⁎C⁎-quad modules and prove that they have universal properties subject to certain operator relations. We also present examples arising from commuting matrices.