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

Abstract

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

48 pages

References in corpus (7)

• Higher rank graph C*-algebras
• C*-algebras of labelled graphs
• $C^{\ast }$-algebras associated with textile dynamical systems
• 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