Characterizing Liminal And Type I Graph C*-Algebras

Details Characterizing Liminal And Type I Graph C*-Algebras Characterizing Liminal And Type I Graph C*-Algebras

2002-11-15

arxiv.org/abs/math/0211241v3

Mathematics

Operator Algebras

20 pages

Scientific paper

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 $\lambda$, there are only finitely many vertices of $\lambda$ that get back to $\lambda$ in an infinite number of ways.

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