K-theory for the simple $C^{\ast }$-algebra of the Fibonacchi Dyck system

Abstract

Let $F$ be the Fibonacci matrix $\left[\begin{array}{ccc}1& 11& 0\end{array}\right]$. The Fibonacci Dyck shift is a subshsystem of the Dyck shift $D_2$ constrained by the matrix $F$. Let ${\mathfrak{L}}^{Ch(D_F)}$ be a $\lambda$-graph system presenting the subshift $D_F$, that is called the Cantor horizon $\lambda$-graph system for $D_F$. We will study the $C^{\ast }$-algebra ${\mathcal{O}}_{{\mathfrak{L}}^{Ch(D_F)}}$ associated with ${\mathfrak{L}}^{Ch(D_F)}$. It is simple purely infinite and generated by four partial isometries with some operator relations. We will compute the K-theory of the $C^{\ast }$-algebra. As a result, the $C^{\ast }$-algebra is simple purely infinite and not semiprojective. Hence it is not stably isomorphic to any Cuntz-Krieger algebra.

18pages