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We give a concrete formula for the unique faithful trace on the finite simplenon-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 ofthe existence of a labeled graph $C^*$-algebra that is not Morita equivalent toany graph $C^*$-algebras. Furthermore, we compute the $K$-groups of$C^*(E_{\mathbb{Z}}, \mathcal{L}, \overline{\mathcal{E}}_{\mathbb{Z}})$ usingthe path structure of the Thue--Morse sequence.

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