I am actually interested in the $C^*$-algebras, so perhaps my question should be: How can you recognize whether a $C^*$-algebra $A$ is isomorphic to $C^*(\Lambda)$ for some (higher-rank) graph $\Lambda$, if $A$ is not presented to you via the Cuntz-Krieger relations?
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$\begingroup$ I'm aware of the work of Drinen, showing that any AF algebra is a graph C*-algebra; I'm asking if anyone knows of more general results along those lines. $\endgroup$– Elizabeth GCommented Aug 29, 2013 at 10:43
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1$\begingroup$ Hi Elizabeth, I think it would be better if the question in the title and the actual question coincided. $\endgroup$– RasmusCommented Aug 30, 2013 at 12:59
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2 Answers
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This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conclude that $A$ is a (higher-rank) graph algebra?"
My answer is: this would require a classification of the class of $C^\ast$-algebras in question together with a description of the range of the classifying invariant on (higher-rank) graph algebras. This problem is open, even for purely infinite Cuntz--Krieger algebras; see Do phantom Cuntz-Krieger algebras exist? (Edit: this case was solved in http://arxiv.org/abs/1511.09463.)
In the simple purely infinite case, one can say the following: a unital UCT Kirchberg algebra $A$ is a graph algebra if and only if $K_1(A)$ is free. This theorem is due to Wojciech Szymanski.
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Here is an incomplete answer, which I hope can be completed by others; therefore, I will make it CW, and invite others to add to it.
A groupoid is the path groupoid of a graph iff it is equivalent as a groupoid to a disjoint union of wedges of circles. Put another way, every groupoid is classified up to equivalence by a list (up to permutations) of groups, namely the fundamental groups of each component; a groupoid is the path groupoid of a graph iff every group on the list is a free group.
So one answer to your question might be: work out what all the fundamental groups are, and whether they are free.
This is not necessarily that easy. Without knowing more, I would expect that it is a very hard question of deciding whether some random group is free.