A dynamical characterization of diagonal-preserving -isomorphisms of graph -algebras

We characterize when there exists a diagonal-preserving -isomorphism between two graph C -algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs E and F and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving -isomorphism between the graph C -algebras C(E) and C(F) .