Partial actions: Restriction and Globalization

Abstract

Partial actions generalize group actions by allowing the domain of the maps induced by each group element to be subspaces rather than the entire space. Often, partial actions arise from restricting a global action to a subspace, which means they are globalizable. It is a good question to ask whether every partial action is globalizable. In the category of sets, every partial action admits a unique globalization. In the topological setting, however, some properties may be lost in the globalization process. That is, a topological partial action on a Hausdorff space may have an enveloping action that is not Hausdorff. Moreover, a special class of partial actions of a finite group, those with the decomposition property, are guaranteed to have a Hausdorff enveloping space.