  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
$R*_{\alpha}^wG$.

Rings and Algebras;Operator Algebras