In this paper, we discuss the existence of nonexpansive retraction onto the set of common fixed points. Assume that $\varphi=\{T_s: s\in S\}$ is an amenable semigroup of nonexpansive mappings on a closed, convex subset $C$ in a reflexive Banach space $E$ such that the set $F(\varphi)$ of common fixed points of $\varphi$ is nonempty. Among other things, it is shown that if either $C$ has normal structure, or the $T_s$'s are affine, then there exists a nonexpansive retraction $P$ from $C$ onto $F(\varphi)$ such that $PT_t=T_tP=P$ for each $t\in S$ and every closed convex $\varphi$-invariant subset of $C$ is also $P$-invariant; in the case that the mappings are affine, $P$ is also affine, and $Px \in \overline{co}\{ T_tx: t\in S\}$ for each $x\in C$, and it is unique regarding the latter property. Our results extend corresponding results of [T. Suzuki, Some remarks on the set of common fixed points of one-parameter semigroups of nonexpansive mappings in Banach spaces with the Opial property, Nonlinear Anal. 58 (2004), 441-458] and [R. E. Bruck, A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53 (1974), 59-71].
