Simplicity of algebras associated to étale groupoids

Jonathan Brown; Lisa Orloff Clark; Cynthia Farthing; Aidan Sims

doi:10.1007/s00233-013-9546-z

Simplicity of algebras associated to étale groupoids

Jonathan Brown
, Lisa Orloff Clark
, Cynthia Farthing
, Aidan Sims

Research output: Contribution to journal › Article › peer-review

112 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	433-452
Number of pages	20
Journal	Semigroup Forum
Volume	88
Issue number	2
DOIs	https://doi.org/10.1007/s00233-013-9546-z
State	Published - 1 Jan 2014
Externally published	Yes

Keywords

Groupoid
Groupoid algebra
Leavitt path algebra

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/s00233-013-9546-z

Cite this

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title = "Simplicity of algebras associated to {\'e}tale groupoids",

abstract = "We prove that the full C *-algebra of a second-countable, Hausdorff, {\'e}tale, 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.",

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author = "Jonathan Brown and Clark, \{Lisa Orloff\} and Cynthia Farthing and Aidan Sims",

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AU - Brown, Jonathan

AU - Clark, Lisa Orloff

AU - Farthing, Cynthia

AU - Sims, Aidan

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Groupoid

KW - Groupoid algebra

KW - Leavitt path algebra

UR - https://www.scopus.com/pages/publications/84897989497

U2 - 10.1007/s00233-013-9546-z

DO - 10.1007/s00233-013-9546-z

M3 - Article

AN - SCOPUS:84897989497

SN - 0037-1912

VL - 88

SP - 433

EP - 452

JO - Semigroup Forum

JF - Semigroup Forum

IS - 2

ER -

Simplicity of algebras associated to étale groupoids

Abstract

Keywords

ASJC Scopus subject areas

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