A dynamical characterization of diagonal-preserving $^{\ast }$ -isomorphisms of graph $C^{\ast }$ -algebras

We characterize when there exists a diagonal-preserving $\ast$ -isomorphism between two graph $C^{\ast }$ -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 $\ast$ -isomorphism between the graph $C^{\ast }$ -algebras $C^{\ast }(E)$ and $C^{\ast }(F)$ .