Tensor Products of Steinberg Algebras

Abstract

We prove that A R (G) ⊗ R A R (H) ∼ = A R (G × 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 ⊗ L 3,R and L 2,R ⊗ L 2,R . Indeed, there are no unexpected diagonalpreserving 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,Z ⊗ L 3,Z ∼ = L 2,Z ⊗ L 2,Z (as * -rings).