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 $C^{\ast }(E)$ is finite, there is a natural isomorphism $C^{\ast }(E)\cong C(E^{\mathrm{\infty }})⋊\mathbb{Z}$, where $E^{\mathrm{\infty }}$ is the infinite path space of $E$ and the action is given by the backwards shift on $E^{\mathrm{\infty }}$. Combining this with a result of Pimsner, we show the properties of being AF-embeddable, quasidiagonal, stably finite, and finite are equivalent for $C^{\ast }(E)$ and can be characterized by a natural “combinatorial” condition on $E$.