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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