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

TL;DR

This paper introduces a new class of $C^*$-algebras derived from Hilbert $C^*$-quad modules associated with $C^*$-textile dynamical systems, extending Pimsner's construction to a two-dimensional setting.

Contribution

It develops a $C^*$-algebra framework from Hilbert $C^*$-quad modules, providing universal properties and examples from commuting matrices, advancing the theory of $C^*$-algebras in dynamical systems.

Findings

Defined a $C^*$-algebra from Hilbert $C^*$-quad modules

Proved universal properties of the constructed $C^*$-algebras

Presented examples from commuting matrices

Abstract

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.