AF labeled graph $C^{\ast }$-algebras

Abstract

It is known that a graph $C^{\ast }$-algebra $C^{\ast }(E)$ 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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebras $C^{\ast }(E)$, this sufficient condition is also a necessary one. Besides, we discuss other equivalent conditions for a graph $C^{\ast }$-algebra to be AF in the setting of labeled graphs and prove that these conditions are not always equivalent by invoking various examples.

20 pages, some new results included

References in corpus (1)

• $C^{\ast }$-algebras of labelled graphs III - $K$-theory computations

Cited by in corpus (1)

• The simplicity of the C*-algebras associated to arbitrary labeled spaces