Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics

Abstract

Let R0 be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of R0, all of which have local units. We show that R0 is maximal commutative in the partial skew group ring R0⋊αG if and only if R0 has the ideal intersection property in R0⋊αG. From this we derive a criterion for simplicity of R0⋊αG in terms of maximal commutativity and G-simplicity of R0. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.