Finiteness Properties of Certain Topological Graph Algebras

Let $E = (E^0, E^1, r, s)$ be a topological graph with no sinks such that $E^0$ and $E^1$ are compact. We show that when $C^*(E)$ is finite, there is a natural isomorphism $C^*(E) \cong C(E^\infty) \rtimes \mathbb{Z}$, where $E^\infty$ is the infinite path space of $E$ and the action is given by the backwards shift on $E^\infty$. Combining this with a result of Pimsner, we show the properties of being AF-embeddable, quasidiagonal, stably finite, and finite are equivalent for $C^*(E)$ and can be characterized by a natural "combinatorial" condition on $E$.