Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics. (Q743912)

Let \(R_0\) be a commutative and associative ring (not necessarily unital), \(G\) a group and \(\alpha\) a partial action of \(G\) on ideals of \(R_0\), all of which have local units. Then, it is proved in the paper under review that \(R_0\) is maximal commutative in the partial skew group ring \(R_0\rtimes_\alpha G\) if and only if \(R_0\) has the ideal intersection property in \(R_0\rtimes_\alpha G\). Then, the authors derive a criterion for simplicity of \(R_0\rtimes_\alpha G\) in terms of maximal commutativity and \(G\)-simplicity of \(R_0\). Two applications of this are then derived in the work: First, a new proof of the simplicity criterion for Leavitt path algebras is given, and second, a new proof of the Cuntz-Krieger uniqueness theorem. Finally, the work also deals with topological dynamics arising from partial actions on clopen subsets of a compact set. The following result is proved in the final section of the work: Let \(\theta=(\{X_t\}_{t\in G},\{h_t\}_{t\in G})\) be a partial action of a group \(G\) on a compact space \(X\) such that for each \(t\in G\), the set \(X_t\) is clopen. Then the partial skew group ring \(C(X)\rtimes_\alpha G\), where \(C(X)\) denotes the continuous complex-valued functions on \(X\), is simple if, and only if, \(\theta\) is topologically free and minimal.