Polish globalization of Polish group partial actions

TL;DR

This paper provides criteria for when a partial action of a Polish group on a Polish space can be extended to a global action that is also Polish, including the existence of a universal globalization.

Contribution

It establishes conditions for the existence of a Polish globalization of partial actions and constructs a universal globalization for Polish group actions.

Findings

Criteria for the existence of a metrizable globalization.

Globalization is Polish if the original spaces are Polish.

Existence of a universal globalization for Polish group partial actions.

Abstract

Let $X$ be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group $G$ on $X.$ If $G$ and $X$ are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.