Finiteness properties of certain topological graph algebras

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 ) ≅ C ( E ∞ ) ⋊ Z , where E ∞ is the infinite path space of E and the action is given by the backwards shift on E ∞ . Combining this with a result of Pimsner, we show the properties of being approximately finite-dimensional-embeddable, quasidiagonal, stably finite, and finite are equivalent for C * ( E ) and can be characterized by a natural ‘combinatorial’ condition on E.