The fixed point property for a class of nonexpansive maps in L\sp\infty(\Omega,\Sigma,\mu)

For a finite and positive measure space $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ and any weakly compact convex subset of $L\text{sp}\mathrm{\infty }(\mathrm{\Omega },\mathrm{\Sigma },mu)$, a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space $C(\mathrm{\Omega })$. An illustrative example is given.