Simplicity of partial skew group rings and maximal commutativity

TL;DR

This paper investigates the conditions under which partial skew group rings are simple and maximally commutative, providing new criteria and proofs for these properties, with applications to Leavitt path algebras and Cuntz-Krieger algebras.

Contribution

It establishes a new equivalence between maximal commutativity and the ideal intersection property in partial skew group rings, and offers novel proofs for simplicity criteria in Leavitt path algebras.

Findings

R0 is maximal commutative iff it has the ideal intersection property in R0*G

Provides a criterion for simplicity based on maximal commutativity and G-simplicity

Offers new proofs for simplicity and the Cuntz-Krieger uniqueness theorem in Leavitt path algebras

Abstract

Let R0 be a commutative associative ring (not necessarily unital), G a group and alpha a partial action by ideals that contain local units. We show that R0 is maximal commutative in the partial skew group ring R0*G if and only if R0 has the ideal intersection property in R0*G. From this we derive a criterion for simplicity of R0*G in terms of maximal commutativity and $G-$simplicity of R0 and apply this to two examples, namely to partial actions by clopen subsets of a compact set and to give a new proof of the simplicity criterion for Leavitt path algebras. A new proof of the Cuntz-Krieger uniqueness theorem for Leavitt path algebras is also provided.