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.

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