  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
$K$.
