Diagonal-preserving isomorphisms of \'etale groupoid algebras

TL;DR

This paper extends the understanding of diagonal-preserving isomorphisms between étale groupoid algebras by broadening coefficient ring conditions and groupoid hypotheses, with applications to Leavitt path algebras.

Contribution

It generalizes existing results by allowing more general coefficient rings and weaker groupoid conditions, requiring only one groupoid to meet certain criteria.

Findings

Extended isomorphism results to broader coefficient rings.

Weakened hypotheses on groupoids for isomorphism detection.

Applications demonstrated in Leavitt path algebras.

Abstract

Work of Jean Renault shows that, for topologically principal \'etale groupoids, a diagonal-preserving isomorphism of reduced $C^*$-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids. Our approach has the additional feature that we only need to impose conditions on one of the two groupoids. Applications are given to Leavitt path algebras.