Finiteness Properties of Certain Topological Graph Algebras

TL;DR

This paper investigates the finiteness properties of topological graph algebras, establishing isomorphisms and equivalences among various finiteness conditions based on the graph's structure.

Contribution

It provides a characterization of when a topological graph algebra is finite, AF-embeddable, quasidiagonal, or stably finite, using a natural combinatorial condition.

Findings

When $C^*(E)$ is finite, it is isomorphic to a crossed product $C(E^ Infty) times Z$.

Finiteness, AF-embeddability, quasidiagonality, and stable finiteness are equivalent properties for $C^*(E)$.

These properties can be characterized by a combinatorial condition on the graph $E$.

Abstract

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 $E$.