$*$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

Abstract: We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two arbitrary countable directed graphs are $$-isomorphic by showing that two Leavitt path algebras over $\mathbb{Z}$ are $$-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 $\mathbb{Z}$ maps the diagonal to the diagonal. Both results hold for slight more general subrings of $\mathbb{C}$ than just $\mathbb{Z}$.