AF labeled graph $C^*$-algebras

TL;DR

This paper investigates when labeled graph $C^*$-algebras are approximately finite dimensional (AF), extending known results from graph $C^*$-algebras and establishing conditions related to loops in labeled spaces.

Contribution

It defines a notion of loops in labeled spaces, provides sufficient conditions for AF-ness, and compares these with known conditions for graph $C^*$-algebras, highlighting their differences.

Findings

AF labeled graph $C^*$-algebras have no loops in the labeled space.

A sufficient condition for AF in labeled spaces is identified.

Conditions for AF in graph $C^*$-algebras are not always equivalent in labeled graph settings.

Abstract

It is known that a graph $C^*$-algebra $C^*(E)$ is approximately finite dimensional (AF) if and only if the graph $E$ has no loops. In this paper we consider the question of when a labeled graph $C^*$-algebra $C^*(E,\CL,\CB)$ is AF. A notion of loop in a labeled space $(E,\CL,\CB)$ is defined when $\CB$ is the smallest one among the accommodating sets that are closed under relative complements and it is proved that if a labeled graph $C^*$-algebra is AF, the labeled space has no loops. A sufficient condition for a labeled space to be associated to AF algebra is also given. For graph $C^*$-algebras $C^*(E)$, this sufficient condition is also a necessary one. Besides, we discuss other equivalent conditions for a graph $C^*$-algebra to be AF in the setting of labeled graphs and prove that these conditions are not always equivalent by invoking various examples.