TENSOR PRODUCTS OF STEINBERG ALGEBRAS

We prove that A(R)(G) circle times(R) A(R)(H) congruent to A(R)(G x H) if G and H are Hausdor ff 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,L-R circle times L-3,L-R and L-2,L- R circle times L-2,L-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 *-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that L-2,L- Z circle times L-3,L- Z not congruent to L-2,L-Z circle times L-2,L- Z (as*-rings).