Cuntz-Krieger algebras associated with Hilbert $C^{\ast }$-quad modules of commuting matrices

Abstract

Let ${\mathcal{O}}_{{\mathcal{H}}_{\kappa }^{A,B}}$ be the $C^{\ast }$-algebra associated with the Hilbert $C^{\ast }$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^{\kappa }$ is transitive, the $C^{\ast }$-algebra ${\mathcal{O}}_{{\mathcal{H}}_{\kappa }^{A,B}}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^{\ast }$-algebra ${\mathcal{O}}_{{\mathcal{H}}_{\kappa }^{[N],[M]}}$ are computed by using the Euclidean algorithm.

19 pages

References in corpus (5)

• $C^{\ast }$-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

Cited by in corpus (1)

• C*-algebras associated with Hilbert C*-quad modules of finite type