TENSOR PRODUCTS OF STEINBERG ALGEBRAS

Abstract

AbstractWe 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$L_{2,R}\otimes L_{2,R}$. In fact, 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$\ast$-rings).