Extrapolation spaces for $C$-semigroups

Let $\{T(t)\}_{t\geq 0}$ be a $C$-semigroup on $X$. We construct an extrapolation space $X_s$, such that $X$ can be continuously densely imbedded in $X_s$, and $\{T_s(t)\}_{t\geq 0}$, the extension of $\{T(t)\}_{t\geq 0}$ to $X_s$, is strongly uniformly continuous and contractive. Using this enlarged space, we give an answer to the question asked in [M. Li, F. L. Huang, Characterizations of contraction $C$-semigroups, Proc. Amer. Math Soc. 126 (1998), 1063-1069] in the negative.