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

TL;DR

This paper studies Cuntz-Krieger algebras derived from Hilbert $C^*$-quad modules associated with commuting matrices, establishing conditions for simplicity and pure infiniteness, and computing their K-groups explicitly.

Contribution

It introduces a new class of Cuntz-Krieger algebras from Hilbert $C^*$-quad modules linked to commuting matrices, and provides explicit K-theory calculations.

Findings

The algebra is simple and purely infinite when the tiling space is transitive.

K-groups are explicitly computed for matrices with entries in {0,1}.

The Euclidean algorithm is used in K-theory calculations.

Abstract

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