Simplicity of algebras associated to étale groupoids

By J. Brown, L. O. Clark, Cynthia Farthing, Aidan Sims. 2012

165 citations recorded by the source corpus.

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 G has totally disconnected unit space, then the complex ∗-algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86a4bdf4a0c1cf84686d5b0dfde9114c486bf4d6

How researchers see this paper

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In Algebra and Number Theory

1. Ben Steinberg
2. Roozbeh Hazrat
3. Enrique Pardo
4. Aidan Sims
5. R. Exel
6. P. Ara
7. Huanhuan Li
8. G. Abrams

In Mathematical Physics

1. Ben Steinberg
2. Aidan Sims
3. Enrique Pardo
4. R. Exel
5. Roozbeh Hazrat
6. P. Ara
7. Huanhuan Li
8. Mark V. Lawson

Citations in the corpus

• \'Etale groupoids and their $C^*$-algebras (2017, methodology): “We therefore obtain the following characterisation of simplicity [8].”
• The tight groupoid of an inverse semigroup (2014, unknown): “4] (see also [2]).”
• Graded Steinberg algebras and their representations (2017, methodology): “For any invariant subset W of the unit space, the free module RW with basis W is a representation of the Steinberg algebra AR(G) [15].”
• Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras (2014, background): “Then we discuss the extension of the simplicity/uniquness results of [4] to arbitrary rings (see the historical discussion below for connections with [5]).”
• Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras (2014, methodology): “Simplicity of OG,E is also discussed by using our description of this algebra as a groupoid C*-algebra and employing results from [4].”