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

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