Ultraweak topology on C$^{*}$-algebras

Question

Let $U$ be a C$^{*}$-algebra of operators acting on a Hilbert space $H$ and let $\overline{U}$ be the weak-operator closure of $U$. I am wondering if the ultraweak topology on the closed unit ball $(U)_{1}$ coincides with the weak-operator topology on $(U)_{1}$ like it does for von Neumann algebras. I think it is true and follows from the fact that the ultraweak topology on $U$ is the relative topology with respect to the ultraweak topology on $\overline{U}$ (this is because every ultraweakly coninuous functional on $U$ extends uniquely to an ultraweakly coninuous functional on $\overline{U}$). Is this reasoning correct?