AF‐embeddability, quasidiagonality and stable finiteness of a $C^{\ast }$‐algebra have been studied by many authors and shown to be equivalent for certain classes of $C^{\ast }$‐algebras. The crossed products $C(X){⋊}_{\sigma }\mathbb{Z}$ (by Pimsner) and $AF{⋊}_{\alpha }\mathbb{Z}$ (by Brown) are such classes, and recently Schfhauser proves the equivalence for $C^{\ast }$‐algebras of compact topological graphs. Clark, an Huef and Sims prove similar results for $k$‐graph algebras. In this paper, we show that this is the case for labeled graph $C^{\ast }$‐algebras $C^{\ast }(E,\mathcal{L})$. Motivated by Schfhauser's result, we also provide another equivalent condition which is easy to check in terms of labeled paths when $(E,\mathcal{L})$ is a labeled graph over a finite alphabet. As a corollary, we have that if $C^{\ast }(E,\mathcal{L})$ is simple, then it is AF‐embeddable if and only if labeled edges have disjoint ranges.

中文翻译：

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可嵌入AF的带标记图C ∗-代数

AF的AF嵌入性，拟对角性和稳定有限性 $C^{\ast }$代数已经被许多作者研究过，并且对于某些类别的代数是等效的。 $C^{\ast }$-代数 划线产品$C（X）{⋊}_{\sigma }\mathbb{ž}$ （由Pimsner撰写）和 $\mathrm{一个}F{⋊}_{\alpha }\mathbb{ž}$ （由布朗出版）属于此类，最近Schfhauser证明了 $C^{\ast }$紧凑拓扑图的代数。Clark，Huef和Sims证明了类似的结果$ķ$图代数。在本文中，我们证明了标记图就是这种情况$C^{\ast }$代数 $C^{\ast }（Ë，\mathcal{大号}）$。根据Schfhauser的结果，我们还提供了另一种等效条件，该条件很容易根据标记路径进行检查$（Ë，\mathcal{大号}）$是有限字母上的标记图。作为推论，如果$C^{\ast }（Ë，\mathcal{大号}）$ 很简单，则当且仅当标记的边缘具有不相交的范围时，它才可嵌入AF。