Unique tracial state on the labeled graph $C^*$-algebra associated to Thue--Morse sequence

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.