On Γ-embeddings and partial actions of function spaces A
Luis A. Martínez-Sánchez, Héctor Pinedo, José L. Vilca-Rodríguez
TL;DR
The paper develops a general framework for extending topological partial actions via Γ-embeddings, proving that any space $Y$ can be Γ-embedded into $C(X,Y)$ when $X$ is compact and constructing a natural induced partial action $\hat{\theta}$ on $C(X,Y)$. It analyzes how enveloping spaces and separation/regularity properties transfer between $Y_G$ and $C(X,Y)_G$, showing that $Y_G$ embeds as an invariant subset of $C(X,Y)_G$ and that ANE/AE properties are preserved across these constructions. The work provides concrete embeddings into the hyperspace $\mathcal{K}(X)$ and the cone $\mathrm{Cone}(X)$, as well as an explicit example demonstrating the equivalence between $C(X,Y)_G$ and $C(X,\mathbb{R})_G$. Overall, it offers a unified method to transport and compare partial actions through function-space and hyperspace embeddings, with implications for globalization and extension problems in topological dynamics.
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 $Γ$- embeddings, and we place particular emphasis on one of them. In particular, we prove that every topological space $Y$ admits a $Γ$-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 $θ$ of a topological group $G$ on $ Y$ naturally induces a partial action $\hatθ$ on $C(X, Y ).$ Throughout the paper, we investigate various relationships between these actions, as well as between their corresponding globalizations and enveloping spaces.