AF labeled graph $C^*$-algebras
J. A. Jeong, E. J. Kang, and S. H. Kim

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 -algebra is approximately finite dimensional (AF) if and only if the graph has no loops. In this paper we consider the question of when a labeled graph -algebra is AF. A notion of loop in a labeled space is defined when is the smallest one among the accommodating sets that are closed under relative complements and it is proved that if a labeled graph -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 -algebras , this sufficient condition is also a necessary one. Besides, we discuss other equivalent conditions for a graph -algebra to be AF in the setting of labeled graphs and prove that these conditions are not always equivalent by invoking various examples.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Noncommutative and Quantum Gravity Theories
