Cuntz-Krieger Algebras Associated with Hilbert $C^*$-Quad Modules of Commuting Matrices

Abstract

Let $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ be the $C^*$-algebra associated with the Hilbert $C^*$-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 $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ is simple and purely infinite. In particular, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{[N],[M]}_{\kappa}}$ are computed by using the Euclidean algorithm.