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A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS

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

Published online by Cambridge University Press:  29 February 2024

BENJAMIN STEINBERG
BENJAMIN STEINBERG*
Affiliation:
Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USA
*
e-mail: bsteinberg@ccny.cuny.edu
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Abstract

Carlsen [‘$\ast $-isomorphism of Leavitt path algebras over $\Bbb Z$’, Adv. Math. 324 (2018), 326–335] showed that any $\ast $-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $\ast $-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

Keywords

étale groupoidgroupoid algebradiagonal-preserving isomorphismLeavitt path algebra

MSC classification

Secondary: 16S99: None of the above, but in this section 22A22: Topological groupoids (including differentiable and Lie groupoids)

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 508 - 513
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author was supported by a Simons Foundation Collaboration Grant, award number 849561, and the Australian Research Council Grant DP230103184.

References

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