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

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