We give a concrete formula for the unique faithful trace on the finite simple non-AF labeled graph $C^*$-algebra $C^*(E_{ \mathbb{ Z}}, \mathcal{ L}, \overline{ \mathcal{ E}}_{ \mathbb{ Z}})$ associated to the Thue--Morse sequence $(E_{ \mathbb{ Z}}, \mathcal{ L})$. Our result provides an alternative proof of the existence of a labeled graph $C^*$-algebra that is not Morita equivalent to any graph $C^*$-algebras. Furthermore, we compute the $K$-groups of $C^*(E_{ \mathbb{ Z}}, \mathcal{ L}, \overline{ \mathcal{ E}}_{ \mathbb{ Z}})$ using the path structure of the Thue--Morse sequence.
Categories: math.OA