Finite simple labeled graph $C^*$-algebras of Cantor minimal subshifts

TL;DR

This paper explores the structure of labeled graph $C^*$-algebras, revealing a new class that are neither AF nor purely infinite, and are isomorphic to crossed products of Cantor minimal subshifts.

Contribution

It demonstrates the existence of simple unital labeled graph $C^*$-algebras that are neither AF nor purely infinite, expanding the understanding of their classification.

Findings

Existence of non-AF, non-purely infinite simple labeled graph $C^*$-algebras.

These algebras are isomorphic to crossed products $C(X)\times_T \mathbb Z$.

They are $A\mathbb T$ algebras with real rank zero and $K_1(\text{algebra})=\mathbb Z$.

Abstract

It is now well known that a simple graph $C^*$-algebra $C^*(E)$ of a directed graph $E$ is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph $C^*$-algebras recently introduced by Bates and Pask as one of the generalizations of graph $C^*$-algebras, and show that there exists a family of simple unital labeled graph $C^*$-algebras which are neither AF nor purely infinite. Actually these algebras are shown to be isomorphic to crossed products $C(X)\times_T \mathbb Z$ where the dynamical systems $(X,T)$ are Cantor minimal subshifts. Then it is an immediate consequence of well known results about this type of crossed products that each labeled graph $C^*$-algebra in the family obtained here is an $A\mathbb T$ algebra with real rank zero and has $\mathbb Z$ as its $K_1$-group.