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

Let $\text{sc}=\{T_s:s\in S\}$ be a representation of a semigroup $S$. First, we prove that the mapping $T_{\mu }$ introduced by a mean on a subspace of $l^{\mathrm{\infty }}(S)$ has many properties of the mappings in the representation $\text{sc}$, in Banach spaces. Then we consider a directed graph and then we define a $Q$-$G$-nonexpansive mapping in locally convex spaces and show that $T_{\mu }$ is a $Q$-$G$-nonexpansive mapping if $T_s$ is a $Q$-$G$-nonexpansive mapping for each $s\in S$. Then we define $Q$-$G$-attractive point of $\text{sc}$ and show if a point $a$ is a $Q$-$G$-attractive point of $\text{sc}$ then $a$ is a $Q$-$G$-attractive point of $T_{\mu }$.