Simplicity of skew group rings of abelian groups

Abstract

Given a group G, a (unital) ring A and a group homomorphism $\sigma : G \to \Aut(A)$, one can construct the skew group ring $A \rtimes_{\sigma} G$. We show that a skew group ring $A \rtimes_{\sigma} G$, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. If G is abelian and A is commutative, then $A \rtimes_{\sigma} G$ is shown to be simple if and only if \sigma is injective and A is G-simple. As an application we show that a transformation group (X,G), where X is a compact Hausdorff space and G is abelian, is minimal and faithful if and only if its associated skew group algebra $C(X) \rtimes_{\sigma} G$ is simple. We also provide an example of a skew group algebra, of an (non-abelian) ICC group, for which the above conclusions fail to hold.