Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of the unit sphere $U'$ of $Y'$. Let $C\subseteq M$ be any weak*-closed convex subset. Is the image $T'(C)$ weak*-closed in $X'$? Or more generally, assuming that $C$ is convex, under which additional conditions on $C$ is the image $T'(C)$ weak*-closed in $X'$?
When you say that $C$ is weak* closed I'm not sure whether you mean as a subset of $Y'$ or in the relative weak* topology on $M$. If the latter, the answer is obviously "no": take $T$ to be the identity map and let $C = M$ be a convex subset of the unit sphere which is not weak* closed. (Examples are easy to find, even in finite dimensions.) If the former, the answer is "yes", for then $C$ is a weak* closed subset of the closed unit ball, hence it is weak* compact, hence its image under a weak* continuous map (which $T'$ is) must also be weak* compact, and hence weak* closed.