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

2017/08/16 by Beuter, Viviane, Gonçalves, Daniel, Öinert, Johan +1
#16S99 #16W22 #16W55 #22A22 #37B05 #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1708.04973

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 \rtimes_πS. Our main result asserts that, when A is commutative, the ring A \rtimes_πS is simple if, and only if, A is a maximal commutative subring of A \rtimes_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 A_R(G) associated with a Hausdorff and ample groupoid G.

Related

• The interplay between Steinberg algebras and partial skew rings
• 'Etale groupoid algebras with coefficients in a sheaf and skew inverse\n semigroup rings
• Graded Steinberg algebras and partial actions
• Simplicity of algebras associated to non-Hausdorff groupoids
• Congruence-simplicity of Steinberg algebras of non-Hausdorff ample groupoids over semifields