AF labeled graph C⁎-algebras

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,L,B) is AF. A notion of loop in a labeled space (E,L,B) is defined when B 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 give rise to an 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.