AF labeled graph C*-algebras

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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,,) is AF. A notion of loop in a labeled space (E,,) 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 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.