# Étale Groupoids and Steinberg Algebras a Concise Introduction

@article{Clark2020taleGA,
title={{\'E}tale Groupoids and Steinberg Algebras a Concise Introduction},
author={Lisa Orloff Clark and Roozbeh Hazrat},
journal={arXiv: Rings and Algebras},
year={2020},
pages={73-101}
}
• Published 2020
• Mathematics
• arXiv: Rings and Algebras
We give a concise introduction to (discrete) algebras arising from etale groupoids (aka Steinberg algebras) and describe their close relationship with groupoid $$C^*$$-algebras. Their connection to partial group rings via inverse semigroups is also explored.
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#### References

SHOWING 1-10 OF 82 REFERENCES
Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras
• B. Steinberg
• Mathematics
• Journal of Pure and Applied Algebra
• 2019
In this paper we give some sufficient and some necessary conditions for an \'etale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroupExpand
Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras
This paper studies simplicity, primitivity and semiprimitivity of algebras associated to \'etale groupoids. Applications to inverse semigroup algebras are presented. The results also recover theExpand
Simplicity of algebras associated to étale groupoids
• Mathematics
• 2012
We prove that the full C∗-algebra of a second-countable, Hausdorff, étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if GExpand
Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras
• Mathematics
• Transactions of the American Mathematical Society
• 2018
We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closedExpand
GRAPH INVERSE SEMIGROUPS, GROUPOIDS AND THEIR C -ALGEBRAS
We develop a theory of graph C -algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting aExpand
Diagonal-preserving graded isomorphisms of Steinberg algebras
• Mathematics
• 2016
We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use thisExpand
Uniqueness Theorems for Steinberg Algebras
• Mathematics
• 2014
We prove Cuntz-Krieger and graded uniqueness theorems for Steinberg algebras. We also show that a Steinberg algebra is basically simple if and only if its associated groupoid is both effective andExpand
Diagonal-preserving isomorphisms of étale groupoid algebras
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 haveExpand