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

TL;DR

This paper provides an explicit formula for the unique faithful trace on a specific labeled graph $C^*$-algebra related to the Thue--Morse sequence, and computes its $K$-groups, highlighting its non-Morita equivalence to standard graph $C^*$-algebras.

Contribution

It offers a concrete trace formula and $K$-theory computations for a novel labeled graph $C^*$-algebra associated with the Thue--Morse sequence, demonstrating its distinctness from traditional graph algebras.

Findings

Explicit formula for the unique faithful trace.

Computed $K$-groups of the algebra.

Established non-Morita equivalence to standard graph $C^*$-algebras.

Abstract

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.