A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS

Carlsen [‘ $\ast $ -isomorphism of Leavitt path algebras over $\Bbb Z$ ’, Adv. Math.324 (2018), 326–335] 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 characterise 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.