Simplicity of Partial Crossed Products

In this article, we consider a twisted partial action $\alpha$ of a group $G$ on a ring $R$ and it is associated partial crossed product $R*_{\alpha}^wG$. We study necessary and sufficient conditions for the commutativity and simplicity of $R*_{\alpha}^wG$. Let $R=C(X)$ the algebra of continuous functions of a topological space $X$ on the complex numbers and $C(X)*_{\alpha}G$ the partial skew group ring, where $\alpha$ is a partial action of a topological group $G$ on $C(X)$. We study some topological properties to obtain results on the algebra $C(X)$. Also, we study the simplicity of $C(X)*_{\alpha}G$ using topological properties and the results about the simplicity of partial crossed product obtained for $R*_{\alpha}^wG$.