Tensor products of Steinberg algebras

We prove that $A_R(G) \otimes_R A_R(H) \cong A_R(G \times H)$, if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R} \otimes L_{3,R}$ and $L_{2,R} \otimes L_{2,R}$. Indeed, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $*$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}} \otimes L_{3,\mathbb{Z}} \not \cong L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ (as $*$-rings).