Characterizing Liminal And Type I Graph C*-Algebras

TL;DR

This paper characterizes when graph C*-algebras are liminal or Type I based on specific finiteness and circuit conditions of the underlying directed graph, providing precise graph-theoretic criteria.

Contribution

It establishes exact graph conditions that determine when a graph C*-algebra is liminal or Type I, linking algebraic properties to combinatorial graph features.

Findings

C*-algebra is liminal iff the graph satisfies a specific finiteness condition.

C*-algebra is Type I iff all circuits are terminal or transitory and a finiteness condition on infinite paths holds.

Provides necessary and sufficient graph-theoretic conditions for liminal and Type I classifications.

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