  It is now well known that a simple graph $C^*$-algebra $C^*(E)$ of a directed
graph $E$ is either AF or purely infinite. In this paper, we address the
question of whether this is the case for labeled graph $C^*$-algebras recently
introduced by Bates and Pask as one of the generalizations of graph
$C^*$-algebras, and show that there exists a family of simple unital labeled
graph $C^*$-algebras which are neither AF nor purely infinite. Actually these
algebras are shown to be isomorphic to crossed products $C(X)\times_T \mathbb
Z$ where the dynamical systems $(X,T)$ are Cantor minimal subshifts. Then it is
an immediate consequence of well known results about this type of crossed
products that each labeled graph $C^*$-algebra in the family obtained here is
an $A\mathbb T$ algebra with real rank zero and has $\mathbb Z$ as its
$K_1$-group.
