$C^{\ast }$-algebras associated with textile dynamical systems

Abstract

A $C^{\ast }$-symbolic dynamical system $(\mathcal{A},\rho ,\Sigma )$ is a finite family $\{{\rho }_{\alpha }{\}}_{\alpha \in \Sigma }$ of endomorphisms of a $C^{\ast }$-algebra $\mathcal{A}$ with some conditions. It yields a $C^{\ast }$-algebra ${\mathcal{O}}_{\rho }$ from an associated Hilbert $C^{\ast }$-bimodule. In this paper, we will extend the notion of $C^{\ast }$-symbolic dynamical system to $C^{\ast }$-textile dynamical system $(\mathcal{A},\rho ,\eta ,{\Sigma }^{\rho },{\Sigma }^{\eta },\kappa )$ which consists of two $C^{\ast }$-symbolic dynamical systems $(\mathcal{A},\rho ,{\Sigma }^{\rho })$ and $(\mathcal{A},\eta ,{\Sigma }^{\eta })$ with certain commutation relations $\kappa$ between their endomorphisms $\{{\rho }_{\alpha }{\}}_{\alpha \in {\Sigma }^{\rho }}$ and $\{{\eta }_a{\}}_{a\in {\Sigma }^{\eta }}$. $C^{\ast }$-textile dynamical systems yield two-dimensional subshifts and $C^{\ast }$-algebras ${\mathcal{O}}_{\rho ,\eta }^{\kappa }$. We will study the structure of the algebras ${\mathcal{O}}_{\rho ,\eta }^{\kappa }$ and present its K-theory formulae.

This paper is a revised version, 54 pages

References in corpus (5)

• C*-algebras of labelled graphs
• Periodic 2-graphs arising from subshifts
• Kumjian-Pask algebras of higher-rank graphs
• $C^{\ast }$-algebras and Fell bundles associated to a textile system
• The tiling C*-algebra viewed as a tight inverse semigroup algebra

Cited by in corpus (3)

• Cuntz-Krieger algebras associated with Hilbert $C^{\ast }$-quad modules of commuting matrices
• C*-algebras associated with Hilbert C*-quad modules of finite type
• C*-algebras associated with Hilbert C*-quad modules of C*-textile dynamical systems