In this talk, we investigate the common fixed point property (fpp) for a commutative family of nonexpansive mappings on $\tau$-compact convex sets in normed and Banach spaces, where $\tau$ is a topological vector space topology that is weaker than the norm topology. We do this through analysis of existence of the nonexpansive retractions. As a consequence of the main results we study the existence and structure of the set of common fixed points of commutative families of nonexpansive self-mappings of a nonempty weak* compact (resp. $clm$-compact) convex subset of the James space $J_0$ (resp. $L_1(\mu)$ , where $\mu$ is $\sigma$-finite).
