On subspaces of invariant vectors

Let $X_{\pi}$ be the subspace of fixed vectors for a uniformly bounded representation $\pi$ of a group $G$ on a Banach space $X$. We study the problem of the existence and uniqueness of a subspace $Y$ that complements $X_{\pi}$ in $X$. Similar questions for $G$-invariant complement to $X_{\pi}$ are considered. We prove that every non-amenable discrete group $G$ has a representation with non-complemented $X_{\pi}$ and find some conditions that provide an $G$-invariant complement. A special attention is given to representations on $C(K)$ that arise from an action of $G$ on a metric compact $K$.