AF标号图$C^*$ -代数
AF labeled graph $C^*$-algebras
摘要：
已知图$C^*$ -代数$C^*(E)$是近似有限维的当且仅当图$E$没有环.本文研究了当一个标号图$C^*$ -代数$C^*(E,\CL,\CB)$是AF时的问题.当$\CB$是相对补闭的容纳集中的最小容纳集时，定义了标记空间$(E,\CL,\CB)$中的环的概念，并证明了当一个标号图$C^*$ -代数是AF时，标记空间没有环.还给出了一个标号空间与AF代数相关联的充分条件。对于图$C^*$ -代数$C^*(E)$，这个充分条件也是一个必要条件.此外，我们还讨论了图$C^*$ -代数在标号图的情况下是AF的其它等价条件，并通过实例证明了这些条件并不总是等价的.
英文摘要：
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.
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