A note on projections in \'etale groupoid algebras and diagonal preserving homomorphisms

Carlsen (Adv.~Math, 2018) showed that any $\ast$-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterize the rings considered by Carlsen as precisely those rings for which every $\ast$-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.