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TENSOR PRODUCTS OF STEINBERG ALGEBRAS

Part of: Topological and differentiable algebraic systems Rings and algebras arising under various constructions

Published online by Cambridge University Press:  04 September 2019

SIMON W. RIGBY Open the ORCID record for SIMON W. RIGBY [Opens in a new window]
SIMON W. RIGBY*
Affiliation:
Department of Mathematics: Algebra and Geometry, Ghent University, Belgium e-mail: [email protected]
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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 $L_{2,R}\otimes L_{2,R}$. 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).

Keywords

Steinberg algebrasample groupoidsLeavitt algebrasdiagonal-preserving isomorphisms

MSC classification

Primary: 16S99: None of the above, but in this section
Secondary: 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 22A22: Topological groupoids (including differentiable and Lie groupoids)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 1 , August 2021 , pp. 111 - 126
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by L. O. Clark

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