Characterizing Liminal And Type I Graph C*-Algebras

Abstract

We prove that the C*-algebra of a directed graph $E$ is liminal iff 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 $λ$, there are only finitely many vertices of $λ$ that get back to $λ$ in an infinite number of ways.