Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$.
A fixed point for $G$ is an element $x$ of $C$ such that $g(x)=x$ for any $g\in G$.
What conditions on $G$ assure the existence of a fixed point for $G$?
The only condition which I know is noncontracting (=distal), see Fixed point theory, Granas/Dugundji, page 173. I need other conditions.