Journal of Operator Theory

Volume 52, Issue 2, Fall 2004 pp. 303-323.

Characterizing liminal and type I graph $C^*$-algebras

Authors: Menassie Ephrem
Author institution: Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804

Summary: We prove that the $C^*$-algebra of a directed graph $E$ is liminal if and only if the graph satisfies the finiteness condition: if $p$ is an infinite path or a path ending with a sink or an infinite emitter, and if $v$ is any vertex, then there are only finitely many paths starting with $v$ and ending with a vertex in $p$. Moreover, $C^*(E)$ is type I precisely when the circuits of $E$ are either terminal or transitory, i.e., $E$ has no vertex which is on multiple circuits, and $E$ satisfies the weaker condition: for any infinite path $\lambda$, there are only finitely many vertices of $\lambda$ that get back to $\lambda$ in an infinite number of ways.

Keywords: Directed graph, Cuntz-Krieger algebra, graph algebra

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