On subspaces of invariant vectors

Let X-pi be the subspace of fixed vectors for a uniformly bounded representation it 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 a 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.