A $C^*$-textile dynamical system $({ \cal A}, \rho,\eta,\Sigma^\rho,\Sigma^\eta, \kappa)$ connsists of a unital $C^*$-algebra ${ \cal A}$, two families of endomorphisms ${ \rho_\alpha}_{ \alpha \in \Sigma^\rho}$ and ${ \eta_a}_{ a \in \Sigma^\eta}$ of ${ \cal A}$ and certain commutation relations $\kappa$ among them. It yields a two-dimensional subshift and multi structure of Hilbert $C^*$-bimodules, which we call a Hilbert $C^*$-quad module. We introduce a $C^*$-algebra from the Hilbert $C^*$-quad module as a two-dimensional analogue of Pimsner's construction of $C^*$-algebras from Hilbert $C^*$-bimodules. We study the $C^*$-algebras defined by the Hilbert $C^*$-quad modules and prove that they have universal properties subject to certain operator relations. We also present its examples arising from commuting matrices.
Categories: math.OA math.DS