Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action $\alpha$ of a group $G$ on an associative algebra $\mathcal{A}$, we consider the crossed product $\mathcal{A}\rtimes _\alpha G$. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of $\mathcal{A}\rtimes_\alpha G$ obtained in the context of $C^*$-algebras. In particular, we prove that $\mathcal{A} \rtimes_{\alpha} G$ is associative, provided that $\mathcal{A}$ is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.