Unique tracial state on the labeled graph $C^{\ast }$-algebra associated to Thue--Morse sequence

Abstract

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

arXiv comment17 pages. [V2]: Section 5(K-group computation), Section 6(representation), and one reference have been added. Minor typos have been corrected. To appear in International Journal of Mathematics