# Journal of Operator Theory

Volume 58, Issue 1, Summer 2007 pp. 205-226.

On the simple $C^*$-algebras arising from Dyck systems**Authors**: Kengo Matsumoto

**Author institution:**Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama, 236-0027 Japan

**Summary:**The Dyck shift $D_N$ for $2N$ brackets $( N >1)$ gives rise to a purely infinite simple $C^*$-algebra ${\it O}_{{\frak L}^{Ch(D_N)}}$, that is not stably isomorphic to any Cuntz-Krieger algebra. It is presented as a unique $C^*$-algebra generated by $N$ partial isometries and $N$ isometries subject to certain operator relations. The canonical AF subalgebra ${\it F}_{{\frak L}^{Ch(D_N)}}$ of ${\it O }_{{\frak L}^{Ch(D_N)}}$ has a unique tracial state. For the gauge action on the $C^*$-algebra ${\it O }_{{\frak L}^{Ch(D_N)}}$, a KMS state at inverse temperature $\log \beta$ exists if and only if $\beta = N+1$ . The admitted KMS state is unique. The GNS representation of ${\it O }_{{\frak L}^{Ch(D_N)}}$ by the KMS state yields a factor of type $\text{III}_{{1}/{(N+1)}}$.

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