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 $\ast$-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).