Tensor products of Steinberg algebras

Abstract

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 $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\ncong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$-rings).

References in corpus (5)

• Graded Steinberg algebras and their representations
• Diagonal-preserving graded isomorphisms of Steinberg algebras
• A Groupoid Approach to Discrete Inverse Semigroup Algebras
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$