On {\Gamma}-embeddings and partial actions of function spaces

TL;DR

This paper introduces a new class of topological embeddings called $ ext{ extGamma}$-embeddings, explores their properties, and demonstrates how they facilitate extending partial group actions to function spaces with the compact-open topology.

Contribution

It defines $ ext{ extGamma}$-embeddings via inverse semigroups of homeomorphisms and shows how they enable extending partial actions to spaces of continuous functions.

Findings

Every topological space admits a $ ext{ extGamma}$-embedding into a space of continuous functions.

Partial actions induce corresponding actions on function spaces, linking their globalizations.

Relationships between partial actions and their extensions are systematically analyzed.

Abstract

This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open subsets of a topological space. We describe several embeddings of this type, referred to as $\Gamma$- embeddings, and we place particular emphasis on one of them. In particular, we prove that every topological space $Y$ admits a $\Gamma$-embedding into the space of continuous functions $C(X, Y )$, equipped with the compact-open topology, where $X$ is a compact space. Consequently, any partial action $\theta$ of a topological group $G$ on $ Y$ naturally induces a partial action $\hat\theta$ on $C(X, Y ).$ Throughout the paper, we investigate various relationships between these actions, as well as between their corresponding globalizations and enveloping spaces.