Some properties of the mapping $T_{\mu}$ introduced by a representation in Banach and locally convex spaces

Let $ \mathcal{S}=\{T_{s}:s\in S\} $ be a representation of a semigroup $S$. We show that the mapping $T_{\mu}$ introduced by a mean on a subspace of $l^{\infty}(S)$ inherits some properties of $\mathcal{S}$ in Banach spaces and locally convex spaces. The notions of $Q$-$G$-nonexpansive mapping and $Q$-$G$-attractive point in locally convex spaces are introduced. We prove that $T_{\mu}$ is a $Q$-$G$-nonexpansive mapping when $T_{s}$ is $Q$-$G$-nonexpansive mapping for each $s\in S$ and a point in a locally convex space is $Q$-$G$-attractive point of $T_{\mu}$ if it is a $Q$-$G$-attractive point of $ \mathcal{S}$.