Cuntz-Krieger algebras associated with Hilbert $C^*$-quad modules of commuting matrices

Let ${\cal O}_{{\cal H}^{A,B}_κ}$ 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 $X_{A,B}^κ$ is transitive, the $C^*$-algebra ${\cal O}_{{\cal H}^{A,B}_κ}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra ${\cal O}_{{\cal H}^{[N],[M]}_κ}$ are computed by using the Euclidean algorithm.

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