On Γ-embeddings and partial actions of function spaces

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.