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Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

  • Viviane Beuter EMAIL logo , Daniel Gonçalves ORCID logo , Johan Öinert ORCID logo and Danilo Royer ORCID logo
Published/Copyright: November 22, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 3

Abstract

Given a partial action π of an inverse semigroup S on a ring 𝒜, one may construct its associated skew inverse semigroup ring 𝒜⋊πS. Our main result asserts that, when 𝒜 is commutative, the ring 𝒜⋊πS is simple if, and only if, 𝒜 is a maximal commutative subring of 𝒜⋊πS and 𝒜 is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra AR⁢(𝒢) associated with a Hausdorff and ample groupoid 𝒢.

Keywords: Skew inverse semigroup ring; Steinberg algebra; partial action; inverse semigroup
MSC 2010: 16S99; 16W22; 16W55; 22A22; 37B05

Communicated by Karl-Hermann Neeb

Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award Identifier / Grant number: 304487/2017-1

Funding statement: The second author was partially supported by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico, under grant number 304487/2017-1.

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Received: 2018-07-06
Revised: 2018-10-17
Published Online: 2018-11-22
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
https://doi.org/10.1515/forum-2018-0160
Keywords for this article
Skew inverse semigroup ring; Steinberg algebra; partial action; inverse semigroup

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
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