⁎-isomorphism of Leavitt path algebras over Z

We characterise when the Leavitt path algebras over Z of two arbitrary countable directed graphs are ⁎-isomorphic by showing that two Leavitt path algebras over Z are ⁎-isomorphic if and only if the corresponding graph groupoids are isomorphic (if and only if there is a diagonal preserving isomorphism between the corresponding graph C⁎-algebras). We also prove that any ⁎-homomorphism between two Leavitt path algebras over Z maps the diagonal to the diagonal. Both results hold for more general subrings of C than just Z.